Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-12-07T13:18:53.611Z Has data issue: false hasContentIssue false
  • English
  • Français

A two dimensional lattice of knots by C2m-moves

Published online by Cambridge University Press:  01 March 2013

SUMIKO HORIUCHI  and
YOSHIYUKI OHYAMA
SUMIKO HORIUCHI
Affiliation:
Department of Mathematics, Tokyo Woman's Christian University, 2-6-1, Zempukuji, Suginami-ku, Tokyo, 167-8585, Japan. e-mail: ohyama@lab.twcu.ac.jp
YOSHIYUKI OHYAMA
Affiliation:
Department of Mathematics, Tokyo Woman's Christian University, 2-6-1, Zempukuji, Suginami-ku, Tokyo, 167-8585, Japan. e-mail: ohyama@lab.twcu.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a local move on a knot diagram, where we denote the local move by λ. If two knots K1 and K2 are transformed into each other by a finite sequence of λ-moves, the λ-distance between K1 and K2 is the minimum number of times of λ-moves needed to transform K1 into K2. A λ-distance satisfies the axioms of distance. A two dimensional lattice of knots by λ-moves is the two dimensional lattice graph which satisfies the following: the vertex set consists of oriented knots and for any two vertices K1 and K2, the distance on the graph from K1 to K2 coincides with the λ-distance between K1 and K2, where the distance on the graph means the number of edges of the shortest path which connects the two knots. Local moves called Cn-moves are closely related to Vassiliev invariants. In this paper, we show that for any given knot K, there is a two dimensional lattice of knots by C2m-moves with the vertex K.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 155 , Issue 1 , July 2013 , pp. 39 - 46
Copyright
Copyright © Cambridge Philosophical Society 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Bar–Natan, D.On the Vassiliev knot invariants. Topology 34 (1995), 423–472.CrossRefGoogle Scholar
[2]Birman, J. S.New points of view in knot theory. Bull. Amer. Math. Soc.(N.S.) 28 (1993), 253–287.CrossRefGoogle Scholar
[3]Birman, J. S. and Lin, X.-S.Knot polynomials and Vassiliev's invariants. Invent. Math. 111 (1993), 225–270.CrossRefGoogle Scholar
[4]Gusarov, M. N.Variations of knotted graphs. Geometric techniques of n-equivalence, Algebra i Analiz 12 (2000), 79–125 (in Russian). English translation: St. Petersburg Math. J. 12 (2001), 569–604.Google Scholar
[5]Habiro, K.Aru musubime no kyokusyo sousa no zoku ni tuite (in Japanese). Master thesis University of Tokyo. (1994).Google Scholar
[6]Habiro, K.Claspers and finite type invariants of links. Geom. Topol. 4 (2000), 1–83.CrossRefGoogle Scholar
[7]Hirasawa, M. and Uchida, Y.The Gordian complex of knots. J. Knot Theory Ramifications 11 (3) (2002), 363–368.CrossRefGoogle Scholar
[8]Horiuchi, S.The Jacobi diagram for a Cn-move and the HOMFLY polynomial. J. Knot Theory Ramifications 16 (2) (2007), 227–242.CrossRefGoogle Scholar
[9]Ichihara, K. and Jong, I. D. Gromov hyperbolicity and a variation of the Gordian complex, preprint.Google Scholar
[10]Kanenobu, T. and Miyazawa, Y.HOMFLY polynomials as Vassiliev link invariants. In Knot Theory. Banach Center Publications 42 (Jones, V. F. R., Kania-Bartoszyńska, J., Przytycki, J. H., Traczyk, P. and Turaev, V. G., eds.) (Institute of Mathematics, Polish Academy of Science, Warszawa, 1998), 165–185.Google Scholar
[11]Matveev, S.Generalized surgeries of three-dimensional manifolds and representations of homology sphere. Mat. Zametki 42 (1987), 268–278, 345 (in Russian); English translation, Math. Notes 42 (1987), 651–656.Google Scholar
[12]Murakami, H. and Nakanishi, Y.On a certain move generating link-homology. Math. Ann. 284 (1989), 75–89.CrossRefGoogle Scholar
[13]Nakanishi, Y. and Ohyama, Y.Local moves and Gordian complex. J. Knot Theory Ramifications 15 (9) (2006), 1215–1224.CrossRefGoogle Scholar
[14]Ng, K. Y. and Stanford, T.On Gusarov's groups of knots. Math. Proc. Camb. Phil. Soc. 126 (1999), 63–76.CrossRefGoogle Scholar
[15]Ohyama, Y.The Ck-Gordian complex of knots. J. Knot Theory Ramifications 15 (2006), 73–80.CrossRefGoogle Scholar
[16]Ohyama, Y., Taniyama, K. and Yamada, S.Realization of Vassiliev invariants by unknotting number one knots. Tokyo J. Math. 25 (1) (2002), 17–31.CrossRefGoogle Scholar
[17]Ohyama, Y. and Tsukamoto, T.On Habiro's Cn-moves and Vassiliev invariants of order n. J. Knot Theory Ramifications 8 (1) (1999), 15–23.CrossRefGoogle Scholar
[18]Ohyama, Y. and Yamada, H.A Cn-move for a knot and the coefficients of the Conway polynomial. J. Knot Theory Ramifications 17 (7) (2008), 771–785.CrossRefGoogle Scholar
[19]Taniyama, K. and Yasuhara, A.Band description of knots and Vassiliev invariants. Math. Proc. Camb. Phil. Soc. 133 (2) (2002), 325–343.CrossRefGoogle Scholar
[20]Vassiliev, V. A.Cohomology of knot spaces. In Theory of Singularities and its Applications, ed. Arnold, V. I. Adv. Soviet Math., Vol.1 (American Mathematical Society, 1990), pp. 23–69.CrossRefGoogle Scholar