Multi-Skein Invariants for Welded and Extended Welded Knots and Links

TL;DR

This paper introduces a new class of invariants for welded and extended welded knots and links using multi-skein relations, expanding the tools available for analyzing these generalized knot theories.

Contribution

It develops a skein-theoretic approach to construct invariants for welded and extended welded knots, with conditions ensuring invariance under extended Reidemeister moves.

Findings

New multi-skein invariants for welded knots

Conditions for invariance under extended Reidemeister moves

Extension of virtual knot invariants to welded and extended welded knots

Abstract

The theory of welded and extended welded knots is a generalization of classical knot theory. Welded (resp. extended welded) knot diagrams include virtual crossings (resp. virtual crossings and wen marks) and are equivalent under an extended set of Reidemeister-type moves. We present a new class of invariants for welded and extended welded knots and links using a multi-skein relation, following Z. Yang's approach for virtual knots. Using this skein-theoretic approach, we find sufficient conditions on the coefficients to obtain invariance under the extended Reidemeister moves appropriate to welded and extended welded links.