On some moves on links and the Hopf crossing number

2019/08/01 by Mroczkowski, Maciej
#57M25 #57M27 #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1908.00342

Abstract

We consider arrow diagrams of links in S³ and define k-moves on such diagrams, for any k∈\mathbb N. We study the equivalence classes of links in S³ up to k-moves. For k=2, we show that any two knots are equivalent, whereas it is not true for links. We show that the Jones polynomial at a k-th primitive root of unity is unchanged by a k-move, when k is odd. It is multiplied by -1, when k is even. It follows that, for any k≥ 5, there are infinitely many classes of knots modulo k-moves. We use these results to study the Hopf crossing number. In particular, we show that it is unbounded for some families of knots. We also interpret k-moves as some identifications between links in different lens spaces L_p,1.

Related

• Non-triviality of the Jones polynomial and the crossing numbers of amphicheiral knots
• 5-move equivalence classes of links and their algebraic invariants
• The Montesinos-Nakanishi 3-move conjecture for links up to 20 crossings
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• The classification of links up to clasp-pass moves