Invariants of welded virtual knots via crossed module invariants of knotted surfaces

Invariants of Welded Virtual Knots Via Crossed Module Invariants of Knotted Surfaces

We define an invariant of welded virtual knots from each finite crossed module by considering crossed module invariants of ribbon knotted surfaces which are naturally associated with them. We elucidate that the invariants obtained are non-trivial by calculating explicit examples. We define welded virtual graphs and consider invariants of them defined in a similar way. 2000 Mathematics Subject C...

Polynomial invariants of graphs on surfaces and virtual knots

QUATERNIONIC INVARIANTS of VIRTUAL KNOTS and LINKS

In this paper we define and give examples of a family of polynomial invariants of virtual knots and links. They arise by considering certain 2×2 matrices with entries in a possibly non-commutative ring, for example the quaternions. These polynomials are sufficiently powerful to distinguish the Kishino knot from any classical knot, including the unknot. The contents of the paper are as follows

On 2-Dimensional Homotopy Invariants of Complements of Knotted Surfaces

We prove that ifM is a compact 4-manifold provided with a handle decomposition with 1-skeleton X , and if G is a finite crossed module, then the number of crossed module morphisms from the fundamental crossed module Π2(M,X , ∗) = (π1(X , ∗), π2(M,X , ∗), ∂, ⊲) into G can be re-scaled to a manifold invariant IG (i. e. not dependent on the choice of 1-skeleton), a construction similar to David Ye...

Algebraic Invariants of Knots

In the first half, we’ll construct algebraic invariants and go over the classification of high dimensional simple knots. For reference, see Farber, Classification of simple knots. In the second half, we’ll compute invariants and discuss connections to number theory. Time permitting, we’ll go over applications of these invariants. In its most generality, knot theory studies the embeddings of man...