Determinants of rational knots

TL;DR

This paper investigates the Fox coloring invariants of rational knots, providing a method to compute their determinants using finite increasing sequences with specific parity properties.

Contribution

It introduces a novel approach to express the propagation of colors and determinants of rational knots through sequences with defined parity patterns.

Findings

Derived a formula for the determinant of rational knots.

Established a link between coloring propagation and sequence properties.

Provided a new perspective on knot invariants using finite sequences.

Abstract

We study the Fox coloring invariants of rational knots. We express the propagation of the colors down the twists of these knots and ultimately the determinant of them with the help of finite increasing sequences whose terms of even order are even and whose terms of odd order are odd.