Longitudinal Mapping Knot Invariant for SU(2)

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Quandle, knot coloring polynomial, quandle cocycle invariant, SU(2)

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The knot coloring polynomial defined by Eisermann for a finite pointed group is generalized to an infinite pointed group as the longitudinal mapping invariant of a knot. In turn, this can be thought of as a generalization of the quandle 2-cocycle invariant for finite quandles. If the group is a topological group, then this invariant can be thought of as a topological generalization of the 2-cocycle invariant. The longitudinal mapping invariant is based on a meridian–longitude pair in the knot group. We also give an interpretation of the invariant in terms of quandle colorings of a 1-tangle for generalized Alexander quandles without use of a meridian–longitude pair in the knot group. The invariant values are concretely evaluated for the torus knots T(2,n), their mirror images, and the figure eight knot for the group SU(2).

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Journal of Knot Theory and Its Ramifications, v. 27, issue 11, art. 1843014