Given a presentation of a knot group, apply abelianization to the Fox derivatives . The resulting matrix over is an Alexander matrix; suitable maximal minors recover the Alexander polynomial of a knot.
For a reduced knot diagram with two adjacent starred regions, a Kauffman state chooses one corner at every crossing so that every unstarred region contains exactly one chosen corner. Terms in a determinant expansion of a Dehn-presentation Alexander matrix correspond bijectively to these states.
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The Alexander matrix, often used in the study of knot theory, is a specific type of matrix associated with a knot or link. It plays a crucial role in analyzing the topology of knots and can be used to derive the Alexander polynomial, an important invariant of knots. The Alexander matrix is constructed from the following steps: 1. **Representation**: Start with a knot or link diagram. From this diagram, choose a triangular decomposition of the knot/link complement.