Free Knots, Groups, and Finite-Type Invariants

TL;DR

This paper introduces a sequence of new invariants for long virtual knots based on parity, valued in specific groups, which remain unchanged under virtualization, and leads to finite order invariants for knots.

Contribution

It constructs a novel sequence of invariants for long virtual knots using parity and group theory, extending the understanding of knot invariants.

Findings

Invariants are valued in groups represented by grids in (m+1)-space.

All invariants are invariant under virtualization.

Finite order invariants are obtained that do not change under virtualization.

Abstract

Based on a recently introduced by the author notion of {\em parity}, in the present paper we construct a sequence of invariants (indexed by natural numbers $m$) of long virtual knots, valued in certain simply-defined group ${\tilde G}_{m}$ (the Cayley graphs of these groups are represented by grids in the $(m+1)$-space); the conjugacy classes of elements of $G_{m}$ play the role of invariants of {\em compact} virtual knots. By construction, all invariants do not change under {\em virtualization}. Factoring the group algebra of the corresponding group by certain polynomial relations leads to finite order invariants of (long) knots which do not change under {\em virtualization