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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
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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 

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