Kei modules and unoriented link invariants

TL;DR

This paper introduces enhanced invariants for unoriented knots and links using kei modules, which improve upon existing invariants and can detect properties like non-invertibility in virtual knots.

Contribution

It develops a new class of invariants by enhancing kei counting invariants with representations of the kei algebra, providing stronger tools for knot analysis.

Findings

Enhanced invariant is stronger than the unenhanced kei counting invariant.

Example demonstrates the enhanced invariant's increased discriminative power.

Application detects non-invertibility of a virtual knot.

Abstract

We define invariants of unoriented knots and links by enhancing the integral kei counting invariant Phi_X^Z (K) for a finite kei X using representations of the kei algebra, Z_K[X], a quotient of the quandle algebra Z[X] defined by Andruskiewitsch and Grana. We give an example that demonstrates that the enhanced invariant is stronger than the unenhanced kei counting invariant. As an application, we use a quandle module over the Takasaki kei on Z_3 which is not a Z_K[X]-module to detect the non-invertibility of a virtual knot.