A Lattice of Finite-Type Invariants of Virtual Knots

TL;DR

This paper constructs an infinite lattice of groups that generate Kauffman finite-type invariants for long virtual knots, extending known invariants and providing bounds on their complexity.

Contribution

It introduces a new lattice structure combining the Polyak algebra and Manturov's map to generate and extend virtual knot invariants.

Findings

Infinite-dimensional subspaces of invariants at each degree

Multiple inequivalent extensions of the Conway polynomial

Bounds on the rank of each group in the lattice

Abstract

We construct an infinite commutative lattice of groups whose dual spaces give Kauffman finite-type invariants of long virtual knots. The lattice is based "horizontally" upon the Polyak algebra and extended "vertically" using Manturov's functorial map $f$. For each $n$, the $n$-th vertical line in the lattice contains an infinite dimensional subspace of Kauffman finite-type invariants of degree $n$. Moreover, the lattice contains infinitely many inequivalent extensions of the Conway polynomial to long virtual knots, all of which satisfy the same skein relation. Bounds for the rank of each group in the lattice are obtained.