No Access

Geodesic graphs for a homotopy class of virtual knots

Department of Mathematics, School of Arts and Sciences, Tokyo Woman's Christian University, 2-6-1, Zempukuji, Suginami-ku, Tokyo 167-8585, Japan

E-mail Address: horiuchi@cis.twcu.ac.jp

Search for more papers by this author

and

Yoshiyuki Ohyama

Department of Mathematics, School of Arts and Sciences, Tokyo Woman's Christian University, 2-6-1, Zempukuji, Suginami-ku, Tokyo 167-8585, Japan

E-mail Address: ohyama@lab.twcu.ac.jp

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S0218216517500900Cited by:0 (Source: Crossref)

Abstract

We consider a local move, denoted by $λ$ , on knot diagrams or virtual knot diagrams.If two (virtual) knots $K_{1}$ and $K_{2}$ are transformed into each other by a finite sequence of $λ$ moves, the $λ$ distance between $K_{1}$ and $K_{2}$ is the minimum number of times of $λ$ moves needed to transform $K_{1}$ into $K_{2}$ . By $Γ_{λ} (K)$ , we denote the set of all (virtual) knots which can be transformed into a (virtual) knot $K$ by $λ$ moves. A geodesic graph for $Γ_{λ} (K)$ is the graph which satisfies the following: The vertex set consists of (virtual) knots in $Γ_{λ} (K)$ and for any two vertices $K_{1}$ and $K_{2}$ , the distance on the graph from $K_{1}$ to $K_{2}$ coincides with the $λ$ distance between $K_{1}$ and $K_{2}$ . When we consider virtual knots and a crossing change as a local move $λ$ , we show that the $N$ -dimensional lattice graph for any given natural number $N$ and any tree are geodesic graphs for $Γ_{λ} (K)$ .

Keywords:

AMSC: 57M25

References

1. D. Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995) 423–472. Crossref, Web of Science, Google Scholar
2. J. S. Birman and X.-S. Lin, Knot polynomials and Vassiliev's invariants, Invent. Math. 111 (1993) 225–270. Crossref, Web of Science, Google Scholar
3. M. N. Gusarov, 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
4. K. Habiro, Claspers and finite type invariants of links, Geom. Topol. 4 (2000) 1–83. Crossref, Web of Science, Google Scholar
5. M. Hirasawa and Y. Uchida, The Gordian complex of knots, J. Knot Theory Ramifications 11(3) (2002) 363–368. Link, Web of Science, Google Scholar
6. S. Horiuchi and Y. Ohyama, A two dimensional lattice of knots by $C_{2 m}$ -moves, Math. Proc. Camb. Phil. Soc. 155 (2013) 39–46. Crossref, Web of Science, Google Scholar
7. S. Horiuchi and Y. Ohyama, A two dimensional lattice of knots by $C_{n}$ -moves, preprint. Google Scholar
8. K. Ichihara and I. D. Jong, Gromov hyperbolicity and a variation of the Gordian complex, Proc. Japan Acad. Ser. A Math. Soc. 87(2) (2011) 17–21. Crossref, Google Scholar
9. L. H. Kauffman, Virtual knot theory, European J. Combin. 20(7) (1999) 693–691. Crossref, Web of Science, Google Scholar
10. Y. Nakanishi and Y. Ohyama, Local moves and Gordian complex, J. Knot Theory Ramifications, 15(9) (2006) 1215–1224. Link, Web of Science, Google Scholar
11. S. Satoh and K. Taniguchi, The writhes of a virtual knot, Fund. Math. 2014, 327–342. Google Scholar
12. V. A. Vassiliev, Cohomology of knot spaces, in Theory of Singularities and its Applications, ed. V. I. Arnold, Advanced. Soviet Mathematics (Springer, 1990). Crossref, Google Scholar