Abstract
We introduce a quandle version of the normalized (twisted) Alexander polynomial, which is an invariant of a pair of an oriented link and a quandle representation. The invariant can be constructed by fixing each Alexander pair, and we find various invariants in our framework, which include the quandle cocycle invariant and the normalized (twisted) Alexander polynomial of a knot. In this paper, we develop the theory of normalization with row and column relations of matrices. The theory works for several row and column relations, although the twisted Alexander polynomial is defined with one column relation. We give a formula of our invariant for the mirror image of an oriented link, which explains why the Alexander polynomial fails to detect the chirality of knots and why the quandle cocycle invariant effectively detects it from a unified point of view. We also show that cohomologous Alexander pairs yield the same invariant.
Published Version
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