The colored HOMFLY-PT polynomials of the trefoil knot, the figure-eight knot and twist knots

TL;DR

This paper rigorously proves explicit formulas for the colored HOMFLY-PT polynomials of the trefoil, figure-eight, and twist knots, expressing them as sums that facilitate their computation and analysis.

Contribution

It provides the first rigorous derivation of these polynomials in sum form for these specific knots, advancing knot invariant computations.

Findings

Explicit sum formulas for trefoil and figure-eight knots

Double sum formula for twist knots

Enhanced understanding of knot invariants

Abstract

We give a rigorous proof of the colored HOMFLY-PT polynomials of the trefoil knot, the figure-eight knot and twist knots. For the trefoil knot and the figure-eight knot, it is expressed by a single sum, and for a twist knot, it is expressed by a double sum.