Updated on 2022/01/28

写真a

 
YASUHARA, Akira
 
Affiliation
Faculty of Commerce, School of Commerce
Job title
Professor

Research Institute

  • 2018
    -
     

    産業経営研究所   兼任研究所員

Education

  • 1991.04
    -
    1993.03

    Waseda University   Doctoral Program, Graduate School of Science and Engineering   Department of Mathematics  

  • 1989.04
    -
    1991.03

    Waseda University   Master Program, Graduate School of Science and Engineering   Department of Mathematics  

  • 1985.04
    -
    1989.03

    Waseda University   School of Education   Department of Mathematics  

Degree

  • 早稲田大学   博士(理学)

  • Waseda University   Ph. D

Research Experience

  • 2018.04
    -
     

    Waseda University   Faculty of Commerce   Professor

  • 2016.04
    -
    2018.03

    Tsuda University   College of Liberal Arts   Professor

  • 2011.07
    -
    2016.03

    Tokyo Gakugei University   Faculty of Education   Professor

  • 2007.04
    -
    2011.06

    Tokyo Gakugei University   Faculty of Education   Associate Professor

  • 1999.07
    -
    2007.03

    Tokyo Gakugei University   Faculty of Education   Associate Professor

  • 1996.04
    -
    1999.06

    Tokyo Gakugei University   Faculty of Education   Assistant Professor

  • 1993.04
    -
    1996.03

    Tokyo Denki University   Department of Mathematical Sciences   Research Assistant

▼display all

 

Research Areas

  • Geometry

Research Interests

  • 位相幾何学

Papers

  • Generalized virtualization on welded links

    Haruko A. MIYAZAWA, Kodai WADA, Akira YASUHARA

    Journal of the Mathematical Society of Japan   72 ( 3 ) 923 - 944  2020.07  [Refereed]

    DOI

  • Burnside groups and n-moves for links

    Haruko A. Miyazawa, Kodai Wada, Akira Yasuhara

    Proceedings of the American Mathematical Society   147 ( 8 ) 3595 - 3602  2019.08  [Refereed]

    DOI

  • Arrow calculus for welded and classical links

    Jean-Baptiste Meilhan, Akira Yasuhara

    Algebraic & Geometric Topology   19   397 - 456  2019.02  [Refereed]

    DOI

  • The pass move is an unknotting operation for welded knots

    Takuji Nakamura, Yasutaka Nakanishi, Shin Satoh, Akira Yasuhara

      247   9 - 19  2018.09  [Refereed]

  • Link invariants derived from multiplexing of crossings

    Haruko Aida Miyazawa, Kodai Wada, Akira Yasuhara

      18   2497 - 2507  2018.04  [Refereed]

  • Linking invariants of even virtual links

    Haruko A. Miyazawa, Kodai Wada, Akira Yasuhara

    JOURNAL OF KNOT THEORY AND ITS RAMIFICATIONS   26 ( 12 ) 1750072-1 - 12  2017.10  [Refereed]

     View Summary

    A virtual link diagram is even if the virtual crossings divide each component into an even number of arcs. The set of even virtual link diagrams is closed under classical and virtual Reidemeister moves, and it contains the set of classical link diagrams. For an even virtual link diagram, we define a certain linking invariant which is similar to the linking number. In contrast to the usual linking number, our linking invariant is not preserved under the forbidden moves. In particular, for two fused isotopic even virtual link diagrams, the difference between the linking invariants of them gives a lower bound of the minimal number of forbidden moves needed to deform one into the other. Moreover, we give an example which shows that the lower bound is best possible.

    DOI

  • Milnor invariants of covering links

    Natsuka Kobayashi, Kodai Wada, Akira Yasuhara

    TOPOLOGY AND ITS APPLICATIONS   224   60 - 72  2017.06  [Refereed]

     View Summary

    We consider Milnor invariants for certain covering links as a generalization of covering linkage invariants formulated by R. Hartley and K. Murasugi. A set of Milnor invariants of covering links is a cobordism invariant of a link, and this invariant can detect some links undetected by the ordinary Milnor invariants. Moreover, for a Brunnian link L, the first non-vanishing Milnor invariant of L is modulo-2 congruent to a sum of Milnor invariants of covering links. As a consequence, a sum of linking numbers of 'iterated' covering links gives the first non-vanishing Milnor invariant of L modulo 2. (C) 2017 Elsevier B.V. All rights reserved.

    DOI

  • Milnor invariants of clover links

    Kodai Wada, Akira Yasuhara

    INTERNATIONAL JOURNAL OF MATHEMATICS   27 ( 13 ) 1650108-1 - 17  2016.12  [Refereed]

     View Summary

    Levine introduced clover links to investigate the indeterminacy of Milnor invariants of links. He proved that for a clover link, Milnor numbers of length up to 2k + 1 are well-defined if those of length <= k vanish, and that Milnor numbers of length at least 2k + 2 are not well-defined if those of length k + 1 survive. For a clover link c with vanishing Milnor numbers of length <= k, we show that the Milnor number mu(c)(I) for a sequence I is well-defined by taking modulo the greatest common divisor of the mu(c)(J)' s, where J is any proper subsequence of I obtained by removing at least k + 1 indices. Moreover, if I is a non-repeated sequence of length 2k + 2, the possible range of mu(c)(I) is given explicitly. As an application, we give an edge-homotopy classification of 4-clover links.

    DOI

  • Milnor invariants of length 2k+2 for links with vanishing Milnor invariants of length <= k

    Yuka Kotorii, Akira Yasuhara

    TOPOLOGY AND ITS APPLICATIONS   184   87 - 100  2015.04  [Refereed]

     View Summary

    J.-B. Meilhan and the second author showed that any Milnor (mu) over bar -invariant of length between 3 and 2k + 1 can be represented as a combination of HOMFLYPT polynomial of knots obtained by certain band sum of the link components, if all A-invariants of length <= k vanish. They also showed that their formula does not hold for length 2k + 2. In this paper, we improve their formula to give the (mu) over bar -invariants of length 2k + 2 by adding correction terms. The correction terms can be given by a combination of HOMFLYPT polynomial of knots determined bSr (mu) over bar -invariants of length k + 1. In particular, for any 4-component link the (mu) over bar -invariants of length 4 are given by our formula, since all (mu) over bar -invariants of length 1 vanish. (C) 2015 Elsevier B.V. All rights reserved.

    DOI

  • AN ELEMENTARY PROOF FOR THAT ALL UNORIENTED SPANNING SURFACES OF A LINK ARE RELATED BY ATTACHING/DELETING TUBES AND MOBIUS BANDS

    Akira Yasuhara

    JOURNAL OF KNOT THEORY AND ITS RAMIFICATIONS   23 ( 1 ) 1450004, 1 - 5  2014.01  [Refereed]

     View Summary

    Gordon and Litherland showed that all compact, unoriented, possibly non-orientable surfaces in S-3 bounded by a link are related by attaching/deleting tubes and half twisted bands. In this note we give an elementary proof for this result.

    DOI

  • Abelian quotients of the string link monoid

    Jean-Baptiste Meilhan, Akira Yasuhara

    ALGEBRAIC AND GEOMETRIC TOPOLOGY   14 ( 3 ) 1461 - 1488  2014  [Refereed]

     View Summary

    The set SL(n) of n-string links has a monoid structure, given by the stacking product. When considered up to concordance, SL(n) becomes a group, which is known to be abelian only if n = 1. In this paper, we consider two families of equivalence relations which endow SL(n) with a group structure, namely the C-k-equivalence introduced by Habiro in connection with finite-type invariants theory, and the C-k-concordance, which is generated by C-k-equivalence and concordance. We investigate under which condition these groups are abelian, and give applications to finite-type invariants.

    DOI

  • Local moves for links with common sublinks

    Jean-Baptiste Meilhan, Eri Seida, Akira Yasuhara

    Topology and its Applications   160   836 - 843  2013  [Refereed]

  • ON NON-SELF LOCAL MOVES

    Yasutaka Nakanishi, Tetsuo Shibuya, Tatsuya Tsukamoto, Akira Yasuhara

    JOURNAL OF KNOT THEORY AND ITS RAMIFICATIONS   21 ( 4 ) 1250055 1 - 9  2012.04  [Refereed]

     View Summary

    Gusarov and Habiro introduced a C-m move, that is strongly related to Vassiliev invariants. In this note, we study a special kind of C-m move, called a non-self C-m move. We show that two links can be transformed into each other by a finite sequence of non-self C-m moves if and only if (1) the two links can be transformed into each other by a finite sequence of C-m moves, and (2) the knot types of corresponding components coincide.

    DOI

  • Milnor invariants and the HOMFLYPT Polynomial

    Jean-Baptiste Meilhan, Akira Yasuhara

    GEOMETRY & TOPOLOGY   16 ( 2 ) 889 - 917  2012  [Refereed]

     View Summary

    We give formulas expressing Milnor invariants of an n-component link L in the 3-sphere in terms of the HOMFLYPT polynomial as follows. If the Milnor invariant (mu) over bar (L)(J) vanishes for any sequence J with length at most k, then any Milnor (mu) over bar -invariant (mu) over bar (L)(I) with length between 3 and 2 k C 1 can be represented as a combination of HOMFLYPT polynomial of knots obtained from the link by certain band sum operations. In particular, the "first nonvanishing" Milnor invariants can be always represented as such a linear combination.

    DOI

  • Regular homotopic deformation of compact surface with boundary and mapping class group

    Susumu Hirose, Akira Yasuhara

    Journal of Knot Theory and its Ramifications   20 ( 10 ) 1391 - 1396  2011.10  [Refereed]

     View Summary

    A necessary and sufficient algebraic condition for a diffeomorphism over a surface embedded in S3 to be induced by a regular homotopic deformation is discussed, and a formula for the number of signed pass moves needed for this regular homotopy is given. © 2011 World Scientific Publishing Company.

    DOI

  • WHITEHEAD DOUBLE AND MILNOR INVARIANTS

    Jean-Baptiste Meilhan, Akira Yasuhara

    OSAKA JOURNAL OF MATHEMATICS   48 ( 2 ) 371 - 381  2011.06  [Refereed]

     View Summary

    We consider the operation of Whitehead double on a component of a link and study the behavior of Milnor invariants under this operation. We show that this operation turns a link whose Milnor invariants of length <= k are all zero into a link with vanishing Milnor invariants of length <= 2k + 1. and we provide formulae for the first non-vanishing ones. As a consequence, we obtain statements relating the notions of link-homotopy and self Delta-equivalence via the Whitehead double operation. By using our result, we show that a Brunnian link L is link-homotopic to the unlink if and only if the link L with a single component Whitehead doubled is self Delta-equivalent to the unlink.

  • Characterization of finite type string link invariants of degree < 5

    Jean-Baptiste Meilhan, Akira Yasuhara

    MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY   148   439 - 472  2010.05  [Refereed]

     View Summary

    We give a complete set of finite type string link invariants of degree < 5. In addition to Milnor invariants, these include several string link invariants constructed by evaluating knot invariants on certain closures of (cabled) string links. We show that finite type invariants classify string links up to C(k)-moves for k <= 5, which proves, at low degree, a conjecture due to Goussarov and Habiro. We also give a similar classification of string links up to C(k)-moves and concordance for k <= 6.

    DOI

  • Homotopy, Delta-equivalence and concordance for knots in the complement of a trivial link

    Thomas Fleming, Tetsuo Shibuya, Tatsuya Tsukamoto, Akira Yasuhara

    TOPOLOGY AND ITS APPLICATIONS   157 ( 7 ) 1215 - 1227  2010.05  [Refereed]

     View Summary

    Link-homotopy and self Delta-equivalence are equivalence relations on links. It was shown by J. Milnor (resp. the last author) that Milnor invariants determine whether or not a link is link-homotopic (resp. self Delta-equivalent) to a trivial link. We study link-homotopy and self Delta-equivalence on a certain component of a link with fixing the other components, in other words, homotopy and Delta-equivalence of knots in the complement of a certain link. We show that Milnor invariants determine whether a knot in the complement of a trivial link is null-homotopic, and give a sufficient condition for such a knot to be Delta-equivalent to the trivial knot. We also give a sufficient condition for knots in the complements of the trivial knot to be equivalent up to Delta-equivalence and concordance. (C) 2010 Elsevier B.V. All rights reserved.

    DOI

  • An estimation of the Ck-unknotting number for a C_k-trivial link

    Teruhisa Kadokami, Akira Yasuhara

    Kobe Journal of Mathematics   27   35 - 46  2010  [Refereed]

  • SELF DELTA-EQUIVALENCE FOR LINKS WHOSE MILNOR'S ISOTOPY INVARIANTS VANISH

    Akira Yasuhara

    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY   361 ( 9 ) 4721 - 4749  2009.09  [Refereed]

     View Summary

    For all n-component link, Milnor'z isotopy invariants are defined for each multi-index I = i(1) i(2)...i(m) (i(j) is an element of {1,..., n}). Here m is called the length. Let r(I) denote the maximum number of times that any index appears in I. It is known that Milnor invariants with r = 1, i.e., Milnor invariants for all multi-indices I with r(I) = 1, are link-homotopy invariant. N. Habegger and X. S. Lin showed that two string links are link-homotopic if and only if their Milnor invariants with r = 1 coincide. This gives us that a link in S(3) is link-homotopic to a trivial link if and only if all Milnor invariants of the link with r = 1 vanish. Although Milnor invariants with r = 2 are not link-homotopy invariants, T. Fleming and the author showed that Milnor invariants with r <= 2 are self Delta-equivalence invariants. In this paper, we give a self Delta-equivalence classification of the set of n-component links in S(3) whose Milnor invariants with length <= 2n - 1 and r <= 2 vanish. As a corollary, we have that a link is self Delta-equivalent to a trivial link if and only if all Milnor invariants of the link with r <= 2 vanish. This is a geometric characterization for links Whose Milnor invariants with r <= 2 vanish. The chief ingredient in our proof is Habiro's clasper theory. We also give all alternate proof of a link-homotopy classification of string links by using clasper theory.

  • MILNOR'S INVARIANTS AND SELF C-k-EQUIVALENCE

    Thomas Fleming, Akira Yasuhara

    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY   137 ( 2 ) 761 - 770  2009  [Refereed]

     View Summary

    It has long been known that a Milnor invariant with no repeated index is an invariant of link homotopy. We show that Milnor's invariants with repeated indices are invariants not only of isotopy, but also of self C-k-equivalence. Here self C-k-equivalence is a natural generalization of link homotopy based on certain degree k clasper surgeries, which provides a filtration of link homotopy classes.

  • Classification of string links up to self delta-moves and concordance

    Akira Yasuhara

    Algebraic and Geometric Topology   9 ( 1 ) 265 - 275  2009  [Refereed]

     View Summary

    For an n-component string link, the Milnor's concordance invariant is defined for each sequence I = i1i2...im(ij,in {1,...,n}) Let r(I) denote the maximum number of times that any index appears. We show that two string links are equivalent up to self Δ-moves and concordance if and only if their Milnor invariants coincide for all sequences I with r(I) ≤ 2. © 2009 Mathematical Sciences Publishers.

    DOI

  • ON C-n-MOVES FOR LINKS

    Jean-Baptiste Meilhan, Akira Yasuhara

    PACIFIC JOURNAL OF MATHEMATICS   238 ( 1 ) 119 - 143  2008.11  [Refereed]

     View Summary

    A C-n-move is a local move on links defined by Habiro and Goussarov, which can be regarded as a 'higher order crossing change'. We use Milnor invariants with repeating indices to provide several classification results for links up to C-n-moves, under certain restrictions. Namely, we give a classification up to C-4-moves of 2-component links, 3-component Brunnian links and n-component C-3-trivial links. We also classify n-component link-homotopically trivial Brunnian links up to Cn+1-moves.

  • Surfaces in 4-manifolds and their mapping class groups

    Susumu Hirose, Akira Yasuhara

    TOPOLOGY   47 ( 1 ) 41 - 50  2008.01  [Refereed]

     View Summary

    A surface in a smooth 4-manifold is called flexible if, for any diffeomorphism phi on the surface, there is a diffeomorphism on the 4-manifold whose restriction on the surface is phi and which is isotopic to the identity. We investigate a sufficient condition for a smooth 4-manifold M to include flexible knotted surfaces, and introduce a local operation in simply connected 4-manifolds for obtaining a flexible knotted surface from any knotted surface. (C) 2007 Elsevier Ltd. All rights reserved.

    DOI

  • Self Δ-equivalence of links in solid tori in S3

    Tetsuo Shibuya, Akira Yasuhara

    Kobe Journal of Mathematics   25   59 - 64  2008  [Refereed]

  • Boundary links are self delta-equivalent to trivial links

    Tetsuo Shibuya, Akira Yasuhara

    MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY   143   449 - 458  2007.09  [Refereed]

     View Summary

    Self Delta-equivalence is an equivalence relation for links, which is stronger than link-homotopy defined by J. W. Milnor. It was shown that any boundary link is link-homotopic to a trivial link by L. Cervantes and R. A. Fenn and by D. Dimovski independently. In this paper we will show that any boundary link is self Delta-equivalent to a trivial link.

    DOI

  • A factorization of the conway polynomial and covering linkage invariants

    Tatsuya Tsukamoto, Akira Yasuhara

    JOURNAL OF KNOT THEORY AND ITS RAMIFICATIONS   16 ( 5 ) 631 - 640  2007.05  [Refereed]

     View Summary

    Levine showed that the Conway polynomial of a link is a product of two factors: one is the Conway polynomial of a knot which is obtained from the link by banding together the components; and the other is determined by the mu-invariants of a string link with the link as its closure. We give another description of the latter factor: the determinant of a matrix whose entries are linking pairings in the infinite cyclic covering space of the knot complement, which take values in the quotient field of Z[t, t(-1)]. In addition, we give a relation between the Taylor expansion of a linking pairing around t = 1 and derivation on links which is invented by Cochran. In fact, the coefficients of the powers of t-1 will be the linking numbers of certain derived links in S3. Therefore, the first non-vanishing coefficient of the Conway polynomial is determined by the linking numbers in S-3. This generalizes a result of Hoste.

  • Milnors numbers and the self delta classification of 2-string links

    Thomas Fleming, Akira Yasuhara

    Intelligence of Low Dimensional Topology 2006 Knots and Everything Series, World Scientific     25 - 34  2007  [Refereed]

  • C-n-move and its duplicated move of links

    Kazuaki Kobayashi, Tetsuo Shibuya, Akira Yasuhara

    JOURNAL OF KNOT THEORY AND ITS RAMIFICATIONS   15 ( 7 ) 839 - 851  2006.09  [Refereed]

     View Summary

    A local move is a pair of tangles with same end points. Habiro defined a system of local moves, C-n-moves, and showed that two knots have the same Vassiliev invariants of order <= n - 1 if and only if they are transformed into each other by C-n-moves. We define a local move, beta(n)-move, which is obtained from a C-n-move by duplicating a single pair of arcs with same end points. Then we immediately have that a Cn+1-move is realized by a beta(n)-move and that a beta(n)-move is realized by twice C-n-moves. In this note we study the relation between C-n-move and On-move, and in particular, give answers to the following questions: (1) Is a beta(n)-move realized by a finite sequence of Cn+-moves? (2) Is C-n-move realized by a finite sequence of on-moves?

  • Classification of n-component Brunnian links up to C-n-move

    HA Miyazawa, A Yasuhara

    TOPOLOGY AND ITS APPLICATIONS   153 ( 11 ) 1643 - 1650  2006.05  [Refereed]

     View Summary

    We give a classification of n-component links up to C-n-move. In order to prove this classification, we characterize Brunnian links, and have that a Brunnian link is ambient isotopic to a band sum of a trivial link and Milnor's links. (C) 2005 Elsevier B.V. All rights reserved.

    DOI

  • Self delta-equivalence of cobordant links

    Y Nakanishi, T Shibuya, A Yasuhara

    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY   134 ( 8 ) 2465 - 2472  2006  [Refereed]

     View Summary

    Self Delta-equivalence is an equivalence relation for links, which is stronger than the link-homotopy defined by J. Milnor. It is known that cobordant links are link-homotopic and that they are not necessarily self Delta-equivalent. In this paper, we will give a sufficient condition for cobordant links to be self Delta-equivalent.

  • Brunnian local moves of knots and Vassiliev invariants

    Akira Yasuhara

    FUNDAMENTA MATHEMATICAE   190   289 - 297  2006  [Refereed]

     View Summary

    K. Habiro gave a neccesary and sufficient condition for knots to have the same Vassiliev invariants in terms of C-k-moves. In this paper we give another geometric condition in terms of Brunnian local moves. The proof is simple and self-contained.

  • Self C-k-move, quasi self C-k-move and the Conway potential function for links

    T Shibuya, A Yasuhara

    JOURNAL OF KNOT THEORY AND ITS RAMIFICATIONS   13 ( 7 ) 877 - 893  2004.11  [Refereed]

     View Summary

    Nakanishi and Shibuya gave a relation between link homotopy and quasi self delta equivalence. And they also gave a necessary condition for two links to be self delta equivalent by using the multivariable Alexander polynomial. Link homotopy and quasi self delta-equivalence are also called self Cl-equivalence and quasi self C-2-equivalence respectively. In this paper, we generalize their results. In Sec. 1, we give a relation between self C-k-equivalence and quasi self Ck+1-equivalence. In Secs. 2 and 3, we give necessary conditions for two links to be self C-k-equivalent by using the multivariable Conway potential function and the Conway polynomial respectively.

  • Linking numbers in rational homology 3-spheres, cyclic branched covers and infinite cyclic covers

    JH Przytycki, A Yasuhara

    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY   356 ( 9 ) 3669 - 3685  2004  [Refereed]

     View Summary

    We study the linking numbers in a rational homology 3-sphere and in the infinite cyclic cover of the complement of a knot. They take values in Q and in Q(Z[t, t(-1)]), respectively, where Q(Z[t, t(-1)]) denotes the quotient field of Z[ t; t 1]. It is known that the modulo-Z linking number in the rational homology 3-sphere is determined by the linking matrix of the framed link and that the modulo-Z[t, t(-1)] linking number in the infinite cyclic cover of the complement of a knot is determined by the Seifert matrix of the knot. We eliminate 'modulo Z' and 'modulo Z[t, t(-1)](,). When the finite cyclic cover of the 3-sphere branched over a knot is a rational homology 3-sphere, the linking number of a pair in the preimage of a link in the 3-sphere is determined by the Goeritz/Seifert matrix of the knot.

  • Signature of rotors

    MK Dabkowski, M Ishiwata, JH Przytycki, A Yasuhara

    FUNDAMENTA MATHEMATICAE   184   79 - 97  2004  [Refereed]

     View Summary

    Rotors were introduced as a generalization of mutation by Anstee, Przytycki and Rolfsen in 1987. In this paper we show that the Tristram-Levine signature is preserved by orient at ion-preserving rotations. Moreover, we show that any link invariant obtained from the characteristic polynomial of the Goeritz matrix, including the Murasugi-Trotter signature, is not changed by rotations. In 2001, P. Traczyk showed that the Conway polynomials of any pair of orientation-preserving rotants coincide. We show that there is a pair of orientation-reversing rotants with different Conway polynomials.

  • Classification of links up to self pass-move

    T Shibuya, A Yasuhara

    JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN   55 ( 4 ) 939 - 946  2003.10  [Refereed]

     View Summary

    A pass-move and a #-move are local moves on oriented links defined by L. H. Kauffman and H. Murakami respectively. Two links are self pass-equivalent (resp. self #-equivalent) if one can be deformed into the other by pass-moves (resp. #-moves), where none of them can occur between distinct components of the link. These relations are equivalence relations on ordered oriented links and stronger than link-homotopy defined by J. Milnor. We give two complete classifications of links with arbitrarily many components up to self pass-equivalence and up to self #-equivalence respectively. So our classifications give subdivisions of link-homotopy classes.

  • Local moves on spatial graphs and finite type invariants

    K Taniyama, A Yasuhara

    PACIFIC JOURNAL OF MATHEMATICS   211 ( 1 ) 183 - 200  2003.09  [Refereed]

     View Summary

    We de. ne A(k)-moves for embeddings of a finite graph into the 3-sphere for each natural number k. Let A(k)-equivalence denote an equivalence relation generated by A(k)-moves and ambient isotopy. A(k)-equivalence implies A(k)-1-equivalence. Let F be an A(k)-1-equivalence class of the embeddings of a finite graph into the 3-sphere. Let G be the quotient set of F under A(k)-equivalence. We show that the set G forms an abelian group under a certain geometric operation. We define finite type invariants on F of order (n; k). And we show that if any finite type invariant of order (1; k) takes the same value on two elements of F, then they are A(k)-equivalent. A(k)-move is a generalization of C-k-move defined by K. Habiro. Habiro showed that two oriented knots are the same up to C-k-move and ambient isotopy if and only if any Vassiliev invariant of order less than or equal to k - 1 takes the same value on them. The 'if' part does not hold for two-component links. Our result gives a sufficient condition for spatial graphs to be C-k-equivalent.

  • Branched covers of tangles in three-balls

    M Ishiwata, JH Przytycki, A Yasuhara

    CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES   46 ( 3 ) 356 - 364  2003.09  [Refereed]

     View Summary

    We give an algorithm for a surgery description of a p-fold cyclic branched cover of B-3 branched along a tangle. We generalize constructions of Montesinos and Akbulut-Kirby.

  • Symmetry of links and classification of lens spaces

    JH Przytycki, A Yasukhara

    GEOMETRIAE DEDICATA   98 ( 1 ) 57 - 61  2003.04  [Refereed]

     View Summary

    We give a concise proof of a classification of lens spaces up to orientation-preserving homeomorphisms. The chief ingredient in our proof is a study of the Alexander polynomial of 'symmetric' links in S-3.

  • C-k-moves on spatial theta-curves and Vassiliev invariants

    A Yasuhara

    TOPOLOGY AND ITS APPLICATIONS   128 ( 2-3 ) 309 - 324  2003.02  [Refereed]

     View Summary

    The C-k-equivalence is an equivalence relation generated by C-k-moves defined by Habiro. Habiro showed that the set of C-k-equivalence classes of the knots forms an abelian group under the connected sum and it can be classified by the additive Vassiliev invariant of order less than or equal to k - 1. We see that the set of C-k-equivalence classes of the spatial theta-curves forms a group under the vertex connected sum and that if the group is abelian, then it can be classified by the additive Vassiliev invariant of order less than or equal to k - 1. However the group is not necessarily abelian. In fact, we show that it is nonabelian for k greater than or equal to 12. As an easy consequence, we have the set of C-k-equivalence classes of m-string links, which forms a group under the composition, is nonabelian for k greater than or equal to 12 and m greater than or equal to 2. (C) 2002 Elsevier Science B.V. All rights reserved.

  • Clasp-pass moves on knots, links and spatial graphs

    K Taniyama, A Yasuhara

    TOPOLOGY AND ITS APPLICATIONS   122 ( 3 ) 501 - 529  2002.08  [Refereed]

     View Summary

    A clasp-pass move is a local move on oriented links introduced by Habiro in 1993. He showed that two knots are transformed into each other by clasp-pass moves if and only if they have the same second coefficient of the Conway polynomial. We extend his classification to two-component links, three-component links, algebraically split links, and spatial embeddings of a planar graph that does not contain disjoint cycles. These are classified in terms of linking numbers, the second coefficient of the Conway polynomial, the Arf invariant, and the Milnor mu-invariant. (C) 2002 Elsevier Science B.V. All rights reserved.

  • Band description of knots and Vassiliev invariants

    Kouki Taniyama, Akira Yasuhara

    Mathematical Proceedings of the Cambridge Philosophical Society   133 ( 2 ) 325 - 343  2002  [Refereed]

    DOI

  • A characterization of four-genus of knots

    T Shibuya, A Yasuhara

    OSAKA JOURNAL OF MATHEMATICS   38 ( 3 ) 611 - 618  2001.09  [Refereed]

  • Realization of knots and links in a spatial graph

    K Taniyama, A Yasuhara

    TOPOLOGY AND ITS APPLICATIONS   112 ( 1 ) 87 - 109  2001.05  [Refereed]

     View Summary

    For a graph G, let Gamma be either the set Gamma (1) of cycles of G or the set Gamma (2) of pairs of disjoint cycles of G. Suppose that for each gamma is an element of Gamma, an embedding phi (gamma) : gamma --> S-3 is given, A set {phi (gamma) \ gamma is an element of Gamma) is realizable if there is an embedding f:G --> S-3 such that the restriction map f\gamma is ambient isotopic to phi (gamma) for any gamma is an element of Gamma. A graph is adaptable if any set {phi (gamma) \ gamma is an element of Gamma (1)} is realizable. In this paper, we have the following three results:
    (1) For the complete graph K-5 on 5 vertices and the complete bipartite graph K-3,K-3 on 3 + 3 vertices, we give a necessary and sufficient condition for {phi (gamma) \ gamma is an element of Gamma (1)} to be realizable in terms of the second coefficient of the Conway polynomial.
    (2) For a graph in the Petersen family, we give a necessary and sufficient condition for {phi (gamma) \ gamma is an element of Gamma (2)} to be realizable in terms of the linking number.
    (3) The set of non-adaptable graphs all of whose proper miners are adaptable contains eight specified planar graphs. (C) 2001 Elsevier Science B.V. All rights reserved.

  • Torus knot that cannot be untied by twisting

    Mohamed Ait Nouh, Akira Yasuhara

    Revista Matemática Complutense   14   423 - 437  2001  [Refereed]

  • Proper links, algebraically split links and Arf invariant

    T Kadokami, A Yasuhara

    JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN   52 ( 3 ) 591 - 608  2000.07  [Refereed]

     View Summary

    In this paper we study certain kinds of links; proper links, algebraically split links and Z(2)-algebraically split links. These links have 'algebraic' definitions. In fact these are defined in terms of the linking number. We shall give these links certain 'geometric' definitions. By using the geometric definitions, we study the Arf invariants of these links.

  • Four-genus and four-dimensional clasp number of a knot

    H Murakami, A Yasuhara

    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY   128 ( 12 ) 3693 - 3699  2000  [Refereed]

     View Summary

    For a knot K in the 3-sphere, by using the linking form on the first homology group of the double branched cover of the 3-sphere, we investigate some numerical invariants, 4-genus g*(K), nonorientable 4-genus gamma*(K) and 4-dimensional clasp number c*(K), defined from the four-dimensional viewpoint. T. Shibuya gave an inequality g*(K) less than or equal to c*(K), and asked whether the equality holds or not. From our result in this paper, we find that the equality does not hold in general.

  • Null-homologous links in certain 4-manifolds

    A Yasuhara

    JOURNAL OF KNOT THEORY AND ITS RAMIFICATIONS   8 ( 1 ) 115 - 123  1999.02  [Refereed]

     View Summary

    Let M be a 4-manifold with partial derivative M congruent to S-3 and L subset of partial derivative M a link. The link L is null-homologous in M if L bounds a disjoint union of once-punctured, orientable surfaces in M. In a previous paper [1] the author defined null-homologous link in 4-manifolds and gave a necessary and sufficient condition for links to be null-homologous in I-manifolds. By using this condition, we investigate the sets of null-homologous links in punctured CP2, <(CP)over bar>(2), CP2 #<(CP)over bar>(2) and S-2 x S-2.

  • Color invariant for spatial graphs

    Y Ishii, A Yasuhara

    JOURNAL OF KNOT THEORY AND ITS RAMIFICATIONS   6 ( 3 ) 319 - 325  1997.06  [Refereed]

     View Summary

    The method of distinguishing knots and links using the colorability of their diagrams was invented by Ralph Fox [2]. As generalization of this method, we introduce certain method of distinguishing spatial graphs.

  • Generalized #-unknotting operations

    K Miyazaki, A Yasuhara

    JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN   49 ( 1 ) 107 - 123  1997.01  [Refereed]

  • Delta-unknotting operation and adaptability of certain graphs

    Akira Yasuhara

    Proceedings of Knots 96 Tokyo, ed. S. Suzuki, World Scientific Publisher     115 - 121  1997  [Refereed]

  • Link homology in 4-manifolds

    A Yasuhara

    BULLETIN OF THE LONDON MATHEMATICAL SOCIETY   28   409 - 412  1996.07  [Refereed]

     View Summary

    We define link homology in 4-manifolds, and show that it has a close connection to linking numbers and intersection matrices of 4-manifolds. We also define null-homologous links in 4-manifolds. We give a necessary and sufficient condition for links to be null-homologous in 4-manifolds. This condition implies that for any 4-manifold with second Betti number n, there are (n + 2)-component links which are not null-homologous in the 4-manifold.

  • Disk band surface and spatial-graph homology

    A Yasuhara

    TOPOLOGY AND ITS APPLICATIONS   69 ( 2 ) 173 - 191  1996.04  [Refereed]

     View Summary

    Using disk/band surfaces, we investigate spatial-graph homology. A necessary and sufficient condition for spatial embeddings of a graph to be homologous is expressed in terms of disk/band surfaces. By this condition we can decide practically whether two given spatial embeddings are homologous or not. We equip the set of spatial-graph homology classes with an abelian group structure and give a method to calculate this group by using the condition above.

  • Connecting lemmas and representing homology classes of simply connected 4-manifolds

    Akira Yasuhara

    Tokyo Journal of Mathematics   19 ( 1 ) 245 - 261  1996  [Refereed]

     View Summary

    We consider surfaces in simply connected 4-manifolds. We estimate the normal Euler numbers of bounded non-orientable surfaces and consider the problem of representing characteristic homology classes by orientable surfaces. To do so, we develop techniques connecting the above problems for given surfaces with the problems for surfaces with fewer first Betti numbers. © 1996 by the University of Notre Dame. All rights reserved.

    DOI

  • Crosscap number of a knot

    H Murakami, A Yasuhara

    PACIFIC JOURNAL OF MATHEMATICS   171 ( 1 ) 261 - 273  1995.11  [Refereed]

     View Summary

    B. E. Clark defined the crosscap number of a knot to be the minimum number of the first Betti numbers of non-orientable surfaces bounding it. In this paper, we investigate the crosscap numbers of knots. We show that the crosscap number of 7(4) is equal to 3. This gives an affirmative answer to a question given by Clark. In general, the crosscap number is not additive under the connected sum. We give a necessary and sufficient condition for the crosscap number to be additive under the connected sum.

  • Knots that cannot be obtained from the trivial knot by a twisting

    Katsura Miyazaki, Akira Yasuhara

    Contemporary Mathematics   164   139 - 150  1994  [Refereed]

  • On C-distance of knots

    Kouki Taniyama, Akira Yasuhara

    Kobe Journal of Mathematics   11   117 - 127  1994  [Refereed]

  • On slice knots in the complex projective plane

    Akira Yasuhara

    Revista Matemática de la Universidad Complutense de Madrid   5   255 - 276  1992  [Refereed]

  • (2, 15)-TORUS KNOT IS NOT SLICE IN CP2

    A YASUHARA

    PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES   67 ( 10 ) 353 - 355  1991.12  [Refereed]

  • On higher dimensional θ-curves

    Akira Yasuhara

    Kobe Journal of Mathematics   8   191 - 196  1991  [Refereed]

  • Disk/Band Surfaces of Spatial Graphs

    Teruhiko SOMA, Hideyuki SUGAI, Akira YASUHARA

    Tokyo Journal of Mathematics   20 ( 1 )  1997.06  [Refereed]

    DOI

▼display all

Books and Other Publications

  • 大学で学ぶ 微分積分[増補版]

    澤田賢, 田中心, 安原晃, 渡邊展也

    サイエンス社  2017.03

  • 大学で学ぶ線形代数

    澤田賢, 渡邊展也, 安原晃

    サイエンス社  2005.03

  • 大学で学ぶ微分積分

    澤田賢, 渡邊展也, 安原晃

    サイエンス社  2005.01

  • 社会科学の数学演習 --線形代数と微積分--

    澤田賢, 渡邊展也, 安原晃

    朝倉書店  2003.03

  • 社会科学の数学 --線形代数と微積分--

    澤田賢, 渡邊展也, 安原晃

    朝倉書店  2002.04

Research Projects

  • ウェルデッド絡み目の有限型不変量とその図形的解釈

    Project Year :

    2017.04
    -
    2020.03
     

     View Summary

    絡み目の一般化であるウェルデッド絡み目に対して,絡み目の有限型不変量の拡張はGoussarov,Polyak,ViroやBar-Natan,Dancsoによる先行研究が知られている.絡み目の場合,有限型不変量の図形的(幾何的)な解釈の研究手段として,Habiroによるクラスパー理論が知られている.一方,ウェルデッド絡み目の有限型不変量の図形的解釈の研究に関しては,幾つかの試みはあるものの満足のいく結果は得られていない。本研究では,ウェルデッド絡み目の有限型不変量の図形的解釈の研究の為に,Habiroのクラスパー理論のウェルデッド絡み目版と見なせるものを新たに定め,ウェルデッド絡み目の有限型不変量の図形的解釈を与えることを目標とする.研究代表者はグルノーブル大学のMeilhan氏との共同研究で,ウェルデッド(ストリグ)絡み目に対し,有限型不変量を保存する局所変形(これを,Wk-moveと呼ぶ)を定める事に成功した.ここで,定義に用いたクラスパーは,HabiroのCkクラスパーと本質的に異なるため,Wkクラスパーと呼ぶ事にする.今年度は,9月にMeilhan氏を訪問し,また3月にMeilhan氏を招聘することで,研究交流をはかり,本研究の進展に役立てた.今年度は,昨年度得られた結果「ウェルデッド・ストリング絡み目のWk同値類は,群の構造を持つ.」を踏まえて,研究を進め,ウェルデッド・ストリング結び目の群をAlexander多項式を用いて完全に特徴つけることに成功した.また,昨年度の「今後の研究の推進方策」で記述した通り,Wk+1同値なウェルデッド(ストリング)絡み目は,Wk同値であることも証明できた.研究計画書に沿って,ほぼ予想通りに研究が進んでいる.平成31年度も,研究交流・成果発表の為の国内出張(大阪,京都,九州)及び海外出張(グルーノーブル,ワシントン)を行い,本研究の進展に役立てる.Meilhan氏との共同研究で得られたWk-moveは,Wk-tree やW-arrowを用いて定義される.Meilhan氏との共同研究では,Wk-treeやW-arrowの変形に関して,Arrow calculasというものも定義した.これは,Habiro氏のクラスパー理論のウェルデッド版を構築する鍵となるもので,応用範囲も広い.今後は,研究計画通りにGoussarov-Habiro予想のウェルデッド版の解決に取り組む事と並行して,Arrow calculasを用いて新しい問題にも取り組みたい

  • 複素射影平面上の直線配置と絡み目のミルナー不変量の研究

    Project Year :

    2015.10
    -
    2018.03
     

     View Summary

    本研究では,絡み目の不変量のアイデアを複素射影平面CP2内の直線配置の研究に応用して,直線配置の位相型を区別する新たな不変量を発見する事を目的とする.本年度は,主に次に上げる研究成果が得られた.これらの成果は,絡み目の不変量を応用した不変量を用いる事で得られた.<BR>(1)実射影変面内の直線配置に関して,図形的に計算可能な絡み数の構成に成功した.これまでの,絡み数は,その計算をコンピューターに依存しなければならない複雑なものばかりであった.ここで構成した絡み数は,計算が簡易であるというばかりでなく,これまで3例しか知られていなかったザリスキー対に加えて,新たに10例のザリスキー対の構成に成功した.さらに,この結果は,今まで知られていなかった有理ザリスキー対の初めての例でもある.(2)直線配置の絡み数の研究を進め,k-アルタル曲線(k ∈ {3, 4, 5, 6})に対してザリスキー対の存在を示した.(直線配置の研究においては,ザリスキー対と呼ばれる直線配置対の発見は,大変重要である.)(3)上で用いた絡み数は,Shimada k-pletと呼ばれる直線配置を区別する事を示した.これらの2つの論文の結果を通して,絡み数が区別する直線配置の特徴が明らかになった.2つ目の結果を得る為に,これまでの絡み数を,代数的・幾何的に再構築し,代数的な計算方法を備えた分離数と関連付けた.(4)基本群は,直線配置の組合せ的構造では,定まらない事を示した.これは,Falk氏とRandell氏の挙げた問題の解答を与えた事になる.また,ここで用いた証明は,Suciu氏の挙げた問題に否定的解答を与えた.29年度が最終年度であるため、記入しない。29年度が最終年度であるため、記入しない

  • Study on Milnor invariants via finite type invariants

    Project Year :

    2014.04
    -
    2017.03
     

     View Summary

    For an n-component link, the Milnor invariant is defined for any finite sequence of numbers in {1,2,...,n}. It is very important in Knot (Link) Theory to study relations between Milnor invariant and other invariants. In this research, we give relations between Milnor invariant and HOMFLYPT polynomial, and Milnor invarinats and Milnor invariants for covering link (covering Milnor invarinats)

  • Finding new mathematical knot theory using data compression

    Project Year :

    2014.04
    -
    2017.03
     

     View Summary

    Milnor invariant is one of most important notions for classifying link groups in the low-dimensional topology. The invariant is, however, not easy to compute for large links. Thus, in this research, we propose a useful tool for this analysis using the grammar compression that makes us possible to compute Milnor invariant for a large depth not computable so far. We developed such a tool and supplied it to the community of researchers for discovering new knowledge of the link groups

  • Study on the classification of string links up to Cn-concordance

    Project Year :

    2011.04
    -
    2014.03
     

     View Summary

    The set SL(m) of m-string links has a monoid structure, given by the stacking product. When considered up to concordance, SL(m) becomes a group, which is known to be abelian only if m=1. In this study, we consider a new equivalence relation which endows SL(m) with a group structure, namely the Cn-concordance, which is generated by C_n-equivalence, which is introduced by Habiro, and concordance. We give a necessary and sufficient condition for a pair (m,n) that the Cn-concordant group of m-string links is abelian

  • OnMilnorinvariantsandclassificationoflinksunder freeselfCn-equivalence

    Project Year :

    2010.04
    -
    2013.03
     

     View Summary

    We introduced simple ribbon moves, and as their extension, simpleribbon fusions toward a systematic construction of ribbon links inspired by Milnor links.Then we studied the effect if we apply them to links, especially on disconnectivity numbersof link introduced by Goldberg and v-th genus, which is a refinement of general genus. Infact, we showed that simple ribbon fusions (moves) never increase the disconnectivitynumbers of links, and never decrease the v-th genus of links

  • 空間グラフのグラフホモロジー類の分類

     View Summary

    Kauffman, Simon, Wolcott, Zhao氏らの共著論文[Invariants of theta-curves and other graphs in 3-spaceの中で定義された空間グラフのdisk/band-曲面は,谷山氏の論文[Cobordism, homotopy and homolog of graphs in R^3]で空間グラフの同値関係として紹介されたグラフホモロジーと密接な関係があり,このdisk/band-曲面を利用して空間グラフのグラフホモロジー類の簡単な分類方法を求めることが本研究の目的であった.与えられた空間グラフに対して,その空間グラフのdisk/band-曲面全体から得られる行列(ザイフェルト行列と呼ばれる)全体の集合はグラフホモロジーの完全不変量である.つまり,このザイフェルト行列の集合を分類することは,グラフホモロジー類を分類ことと同値である.ところが,このザイフェルト行列の集合は無限個の元からなるうえに有限の表示さえもできないので,このままではいささか扱いにくい.従って私は次の2つの条件に沿ってザイフェルト行列を制限し研究を進めた.1.空間グラフのdisk/band-曲面のトポロジカルタイプを1つに固定する.2.disi/band-曲面の1次元ホモロジー群の生成系を1つ固定する.これらの条件を考慮して得られるザイフェルト行列の部分集合に対して,本研究では次の新しい結果を得ることに成功した.[定理]上で得られた部分集合は,各成分が多変数の整数係数1次式からなる行列1個で表現できる.この結果により,本研究の目標は達成されたことになる.実際,空間グラフのグラフホモロジー類の分類問題は,整数係数の連立1次方程式が整数解をもつか否かに帰着できることがこの結果からわかる

  • 空間グラフホモロジー群とグラフマイナ-との関係についての研究

     View Summary

    3次元空間内に埋め込まれたグラフを空間グラフといい,空間グラフの集合の同値関係にグラフホモロジーと呼ばれるものがある.グラフを固定したとき,この同値関係における同値類の集合は,ある演算のもとで有限生成な自由アーベル群(この群を空間グラフホモロジー群と呼ぶ.)の構造をもつ.この空間グラフホモロジー群のランクと,抽象グラフとしてのグラフの複雑さとの間の関係を調べることが本研究の目的であった.ここではグラフの複雑さをはかる目安として,マイナ-と呼ばれるグラフの半順序関係を用いた.グラフHがグラフGのマイナ-であるとは,HがGから「辺の除去」または「辺の縮約」と呼ばれる操作の繰り返しにより得られることを意味するものであるが,昨年度までの私の研究により次の命題が得られていた.[命題]グラフHがグラフGから辺を除去して得られるならば,Hの空間グラフホモロジー群のランクはGの空間グラフホモロジー群のランク以下になる.従って,「辺の縮約」に関しても「辺の除去」の場合と同様のことを示すことが課題であったが,今回の研究により次の結果が得られた.[命題]グラフHがグラフGから辺を縮約して得られるならば,Hの空間グラフホモロジー群のランクはGの空間グラフホモロジー群のランク以下になる.上の2つの命題を合わせることにより目標である次の結果が得られた.[定理]グラフHがグラフGのマイナ-ならば,Hの空間グラフホモロジー群のランクはGの空間グラフホモロジー群のランク以下になる.また,グラフHがグラフGのマイナ-であっても,Hの空間グラフホモロジー群のランクとG空間グラフホモロジー群のランクが等しくなることがある.本研究では,このようなことが生じる場合の条件付けも行った

  • 空間グラフに含まれる結び目と絡み目の研究

     View Summary

    グラフGのサイクル全体の集合を{c_1,c_2…,c_n}とする.任意の埋め込みf_i:c_i→R^3(i=1,2,…,n)に対して,Gの埋め込みf:G→R^3が存在してf(ci)〃〓f_i(c_i)(i=1,2,…,n)を満たすとき,Gは順応性をもつという.グラフが順応性をもつという性質は,グラフの集合の半順序関係であるグラフマイナーに関して閉じている.すなわち,「グラフHがグラフGのグラフマイナー(G【greater than or equal】Hとかく)であるとき,Gが順応性をもつならばHも順応性をもつ.」が成立する.順応性をもたないグラフGに対し,GのG以外に任意のグラフマイナーが順応性をもつとき,Gは順応性に関する禁止グラフと呼ばれ,禁止グラフ全体の集合を順応性に関する障害集合と呼びΩで表す.この障害集合Ωは,有限集合である事がRobertsonとSeymourの結果からわかり,空集合でないという事が谷山公規氏と私の結果からわかるが,それを決定するまでには至っていない.障害集合を決定することは非常に重要な問題であり,かなりの難問である.本研究ではこの障害集合を決定するということを目標に研究を進めてきた.障害集合Ωを次の2つの部分集合Ω_0=Ω∩(平面的グラフの集合)とΩ_1=Ω\Ω_0に分けると,本橋友江,谷山公規両氏の共同研究の結果と最近の私の研究成果を組み合わせることにより,Ω_1={K_5,K_<3,3>}が成立することがわかる.ここで,K_5は5頂点完全グラフを意味し,K_<3,3>は完全2部グラフを意味する.従って,障害集合Ωの決定はΩ_0を決定すれば解決することになる.本年度は,まず次の予想「D_n(n【greater than or equal】3)をnサイクルの各辺を2重にして得られるグラフとする.このときΩ_0={D_4}が成立する.」を立てて研究を始めた.ところが,この予想に反して,D_4は順応性をもつことがわかった.更に,Ω_0に属する幾つかのグラフが見つかった

  • 空間テータ曲線の局所変形とVassiliev不変量の研究

     View Summary

    結び目・絡み目の局所変形にC_k-moveと呼ばれるものがある.これは葉広氏により定義されたもので,同氏は数年前に次の驚くべき結果「2つの結び目の位数k-1のVassiliev不変量が等しい為の必要十分条件は,それらがC_k-moveで移り合うことである.」を示した.結び目の集合をC_k-moveで割った同値類(これをC_k同値類と呼ぶ)は,結び目の連結和という操作の下で可換群になる.これは葉広氏の結果の証明のキーポイントである.結び目理論で良く扱われる空間グラフの1つに,空間θ曲線と呼ばれるものがある.空間θ曲線に対してもC_k-moveが定義できるのであるが,最近の私の研究により,空間θ曲線のC_k同値類が頂点連結和と呼ばれる操作の下で群構造をもつ事がわかり,位数kの低いところでは,葉広氏の定理と同様な結果が得られることがわかった.結び目のC_k同値類のなす群と空間θ曲線のC_k同値類のなす群の大きな違いは,結び目の方は可換群になるのが明らかなのに対し,空間θ曲線の方は位数kが大きくなると可換か否かの判定が困難になる事が挙げられる.平成13年度の研究成果として次が得られた.[定理]空間θ曲線のC_k同値類のなす群が可換であれば,2つの空間θ曲線の位数k-1のVassiliev不変量が等しい為の必要十分条件は,それらがC_k-moveで移り合うことである.従って,空間θ曲線のC_k同値類のなす群が可換か否かを調べることが重要な課題であった.これに対し,平成14年度の成果として次が得られた.[定理]空間θ曲線のC_k同値類のなす群は,k【less than or equal】4ならば可換であり,k【greater than or equal】12ならば非可換である

  • 局所変形で与えられる結び目の同値類のなす群に関する研究

     View Summary

    結び目の局所変形とは,文字どおり結び目を局所的に変形する操作のことである.この局所変形を用いると結び目の集合に同値関係を定義することができる.結び目の局所変形はいろいろ定義され研究されているが,その中でもCk-move(kは自然数)はVassiliev不変量と密接な関係があることが知られており,多数の研究者が興味を持ち研究がされている.Ck-moveで与えられる結び目の同値類は結び目の連結和の下で可換群(Ck-同値群と呼ぶ)になることが知られている.この群は結び目のVassiliev不変量で定義されるGusarov群と同型であることも知られており,結び目のVassiliev不変量の研究において,Ck-同値群の研究は非常に重要である.Ck-同値群の研究において,次の問題「Ck-同値群(=Gusarov群)は自由加群か?」は現在未解決の難問である.本研究ではこの問題の解決の試みとして,Ck-moveに関連した新しい局所変形βk-moveについて研究を行った.昨年度までの研究では,Ck-同値群とβk-同値群は,k=1,3のときは同型で,k=2のときは同型でないことがわかっていた.本年度は,一般のk>3の場合についてβk-同値群の構造を調べ,Ck-同値群とβk-同値群が同型になることがわかった.この結果により,Ck-同値群とβk-同値群の関係が完全に明らかになった

  • Milnor invariants and Classifications of links up to self-type local moves

     View Summary

    For each multi-index I which consists of 1,...,m, the Milnor invariant μ(I) is defined. Let r(I) be the maximum number of times that any index appears in I. We investigate an equivalence relation, self Cn-equivalence, of links which is a generalized link-homotopy. Here, link-homotopy, which is defined by Milnor, is a well-known equivalence relation of links, and μ(I) is a link-homotopy invariant for any I with r(I)=1. We remark that self Cn-equivalence coincides with link-homotopy if n=1.As a result of during the research of 2006, we have that μ(I) is a self Cn-equivalence invariant if(I)≦n.It is known that, for strip links, Milnor invariants with r(I)=1 give the link-homotopy classification. This lets us think of a question : Do Milnor invariants with r(I)≦n give the self Cn-equivalence classification of string links?During the research of 2007, we have a negative answer to the question above even for n=2. We also have that a link is self C_2-equivalent to the trivial link if and only if its Milnor invariants with r(I)≦2 vanish. So Milnor invariants with r(I)≦2 are strong enough to show that a link is self C_2-equivalent to the trivial link, although they are not enough to classify string links up to self C_2-equivalence

  • Cn-equivalence on string links and Milnor invariants

     View Summary

    For any finite sequence I valued in {1, …, m}, Milnor invariant μ(I) for m-string links are defined. We denote by r(I) the maximal number of times which any index appears in I. We give a geometric characterization for string links which have same Milnor invariants μ(I) for any I with r(I) at most 2. And, for a natural number k which is at most 6, we give a geometric characterization for string links which have same Milnor invariants μ(I) for any sequence I with length at most k

▼display all

Specific Research

  • 絡み目コンコーダンスの研究

    2020  

     View Summary

    3次元球体内に埋め込まれた有限個(n個)の円周の集合を(n成分)絡み目と呼ぶ.2つのn成分絡み目LとL'がリンク・コンコーダントであるとは,LとL’が4次元球体内に埋め込まれたn個の円環の境界になるときをいう.また,この埋め込まれた円環をLからL'へのリンク・コンコーダンスと呼ぶ.LからL'へのリンク・コンコーダンス全体の集合C(L,L')には,曲面リンク・ホモトピーという同値関係を定義することができる.C(L,L')の曲面リンク・ホモトピーによる剰余類全体の集合をC(L,L')/lhで表す.研究代表者は,フランスの数学者Jean-Baptiste Meilhan氏との共同研究で,C(L,L')の各元cに対し,(ミルナー型と呼ばれる)不変量μ(c)を定義し,次の結果を得た: (1)&nbsp;μ(c)は曲面リンク・ホモトピー不変量である. (2)&nbsp;LとL'がスライス絡み目(つまり,L, L'は自明な絡み目にリンク・コンコーダント)の場合,μ(c)は,&nbsp;C (L,L')/ lhの完全分類を与える.

  • ストリング絡み目の4-moveとミルナー不変量に関する研究

    2019  

     View Summary

    研究代表者は,津田塾大学の宮沢治子氏と大阪大学の和田康載氏との共同研究において,ストリング絡み目の2n-move(nは自然数)の研究を行い.論文「Classification of sring links up to 2n-moves and link-homotopy」をまとめ,国際的査読付き雑誌に投稿した.また,ストリング絡み目の拡張であるウェルデッドストリング絡み目において,同様の研究成果を得られたので,論文「Milnor invariants, 2n-moves and V^n-moves for welded string links」にまとめた.これは,国際的査読付き雑誌「Tokyo Journal of Mathematics」に掲載予定である.どちらの論文も,Milnor不変量と2n-moveの関係を調べ,2n-moveによる同値関係より,少し弱い同値関係による(ウェルデッド)ストリング絡み目の分類問題を,nを法としたMilnor不変量を用いて解決した.

  • ウェルデッド絡み目の局所変形に関する研究

    2018  

     View Summary

    研究代表者とJean-Baptiste Meilhan氏は,共同研究「Arrow calculusfor welded and classical links, to appera in Algebraic &amp; Geometric Topology」において,ウェルデッド絡み目のArrowCalculusと呼ばれるものを導入し,ウェルデッド絡み目の局所変形に関する新たな手法を導入する事に成功した.本研究では,新たな局所変形を導入し,Arrow Calculusを用いて研究を行った.具体的には,twist-moveの亜種である,V(n)-moveとVn-moveを新たに定義し,これらによって定められるウェルデッド絡み目の同値関係による完全分類を与えた.twist-moveはプリンストン大学のFox教授の1958年の論文に起源をもつ古典的かつ重要な研究課題であり,今なお研究が続いている.本研究で得られた,twist-moveの亜種の局所変形(V(n)-moveとVn-move)による同値類の完全分類の成果はtwist-moveの今後の研究にも大きく貢献するだろう.&nbsp;&nbsp;

 

Syllabus

▼display all