On usual, virtual and welded knotted objects up to homotopy

Abstract

We consider several classes of knotted objects, namely usual, virtual and welded pure braids and string links, and two equivalence relations on those objects, induced by either self-crossing changes or selfvirtualizations. We provide a number of results which point out the differences between these various notions. The proofs are mainly based on the techniques of Gauss diagram formulae.• P n and SL n stand for the (usual) sets of pure braids and string links on n strands, and the prefix v and w refer to their virtual and welded counterpart, respectively; • the superscripts sv and sc refer respectively to the equivalence relations generated by self-virtualization and self-crossing change.

Generalized virtualization on welded links

“…Each arc of D yields a generator, and each classical crossing gives a relation yx −1 yz −1 , where x and z correspond to the underpasses and y corresponds to the overpass at the crossing. This group Π (2) D is known as a classical link invariant [11,14,8,27]. Remark 6.1.…”

“…Remark 6.1. Let L be an unoriented classical link in the 3-sphere and D a classical diagram of L. M. Wada [27] proved that Π (2) D is isomorphic to the free product of the fundamental group of the double branched cover M (2) L of the 3-sphere branched along L and the infinite cyclic group Z. That is, Π…”

“…We similarly define the associated core group Π (2) D of an unoriented virtual link diagram D by generators and relations as described above. (Note that virtual crossings do not produce any generator or relation.)…”

“…(2) D is preserved by welded Reidemeister moves, and hence we define the associated core group Π (2) L of an unoriented welded link L to be Π (2) D of a diagram D of L. Moreover we have the following. Proposition 6.2.…”

“…In classical knot theory, local moves have played important roles and hence have been studied widely; see for example [20,19,10,1,17,5,6]. Recently, some "classical" local moves, which exchange classical tangle diagrams, have been studied for welded knots and links [2,3,26,21]. In this paper, we will study "non-classical" local moves for welded links.…”