Mutant knots and intersection graphs

Abstract

We prove that if a finite order knot invariant does not distinguish mutant knots, then the corresponding weight system depends on the intersection graph of a chord diagram rather than on the diagram itself. The converse statement is easy and well known. We discuss relationship between our results and certain Lie algebra weight systems.

The transition matroid of a 4-regular graph: An introduction

“…Theorem 2 tells us that if an operation on 4-regular graphs preserves interlacement graphs, then it also preserves transition matroids. In Section 4 we discuss a theorem of Ghier [34], a version of which was subsequently discovered independently by Chmutov and Lando [23]. Ghier's theorem implies that there are two fundamental types of interlacement-preserving operations, associated with edge cuts of size two or four.…”

The space of framed chord diagrams as a Hopf module

“…For example, there exists an order 11 weight system, that distinguishes two mutant knots [11], and thus, by a theorem of Chmutov and Lando (see [3]), can not be obtained from a linear map from . Note, that even putting in the previous Theorem ′ = ℎ ∘ , by a suitable choice of , it is possible to construct framed weight system not determined by the intersection graph only.…”

Parity in knot theory and graph-links

“…Remark 2.11. If we consider symmetric matrices realizable by chord diagrams, then the corresponding elements from G(n) and L(n) are just framed 4-graphs up to mutation; see [38,66]. Our isomorphism υ gives us an equivalence between different ways of definition of a framed 4-graph.…”