5-move equivalence classes of links and their algebraic invariants

Mieczyslaw K. Dabkowski; Makiko Ishiwata; Jozef H. Przytycki

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arXiv · 0712.0985

5-move equivalence classes of links and their algebraic invariants

Mieczyslaw K. Dabkowski·Makiko Ishiwata·Jozef H. Przytycki

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Abstract

We start a systematic analysis of links up to 5-move equivalence. Our motivation is to develop tools which later can be used to study skein modules based on the skein relation being deformation of a 5-move (in an analogous way as the Kauffman skein module is a deformation of a 2-move, i.e. a crossing change). Our main tools are Jones and Kauffman polynomials and the fundamental group of the 2-fold branch cover of S^3 along a link. We use also the fact that a 5-move is a composition of two rational \pm (2,2)-moves (i.e. \pm 5/2-moves) and rational moves can be analyzed using the group of Fox colorings and its non-abelian version, the Burnside group of a link. One curious observation is that links related by one (2,2)-move are not 5-move equivalent. In particular, we partially classify (up to 5-moves) 3-braids, pretzel and Montesinos links, and links up to 9 crossings.

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BibTeX RIS

Mieczyslaw K. Dabkowski, Makiko Ishiwata, Jozef H. Przytycki. 2007-12-06. 5-move equivalence classes of links and their algebraic invariants. https://arxiv.org/abs/0712.0985

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