Homology, Homotopy and Applications

Volume 16 (2014)

Number 1

Kei modules and unoriented link invariants

Pages: 167 – 177

DOI: http://dx.doi.org/10.4310/HHA.2014.v16.n1.a10

Authors

Michael Grier (Department of Mathematical Sciences, Claremont McKenna College, Claremont, California, U.S.A.)

Sam Nelson (Department of Mathematical Sciences, Claremont McKenna College, Claremont, California, U.S.A.)

Abstract

We define invariants of unoriented knots and links by enhancing the integral kei counting invariant $\Phi_X^{\mathbb{Z}}(K)$ for a finite kei $X$ using representations of the kei algebra, $\mathbb{Z}_K[X]$, a quotient of the quandle algebra $\mathbb{Z}[X]$ defined by Andruskiewitsch and Graña. 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 $\mathbb{Z}_3$ which is not a $\mathbb{Z}_K[X]$-module to detect the non-invertibility of a virtual knot.

Keywords

Kei algebra, kei module, involutory quandle, enhancement of counting invariants

2010 Mathematics Subject Classification

57M25, 57M27

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