Rational moves and tangle embeddings: (2,2)-moves as a case study
Abstract
We classify 3-braids up to (2,2)-move equivalence and in particular we show how to adjust the Harikae-Nakanishi-Uchida conjecture so it holds for closed 3-braids. As important steps to classify 3-braids up to (2,2)-move equivalence we prove the conjecture for 2-algebraic links and classify (2,2)-equivalence classes for links up to nine crossings. We analyze the behavior of Kei (involutive quandle) associated to a link under (2,2)-moves. We construct Burnside Kei, Q(m,n), and ask the question, motivated by classical Burnside question: for which values of m and n, is Q(m,n) finite?
Cite
@article{arxiv.math/0501539,
title = {Rational moves and tangle embeddings: (2,2)-moves as a case study},
author = {Mieczyslaw K. Dabkowski and Makiko Ishiwata and Jozef H. Przytycki},
journal= {arXiv preprint arXiv:math/0501539},
year = {2007}
}
Comments
15 pages, 17 figures; this e-print is an English version of the paper which will appear in "Topology of Knots VII" (Proceeding of the Conference Topology of Knots VII, Dec. 23-26, 2004, TWCU), February, 2005, 37-46 (in Japanese)