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# 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

#### 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