Undergraduate Seminar: "An Invariant of p-Colorable Knots"

Abstract: Knot invariants have been essential computational tools for tabulation. A linking number is used to describe the number of times two distinct knots wind around each other. From a p-colored knot, for p prime, we can build a new space called a branched cover, where the knot lifts to one knot of branching index 1 and (p-1)/2 knots of branching index 2. From any pair of these knots, we can compute a linking number, called a dihedral linking number. The set of all dihedral linking numbers associated to a p-colorable knot is the dihedral linking invariant. Perko created a combinatorial algorithm for computing the dihedral linking number for all 3-colorable knots. In this work, we generalize Perko’s algorithm to all p-colorable knots.

Host: Adeli Hutton