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?