Compositions of Knots Using Alexander Polynomial

TL;DR

This paper explores how the Alexander Polynomial can be used to understand the composition of knots, providing new insights into knot determinants and their behavior under knot composition.

Contribution

It introduces techniques to analyze the behavior of knot determinants when composing two knots using Alexander Polynomials, advancing knot theory understanding.

Findings

Knot determinants can be generalized through Alexander Polynomials.

The behavior of knot determinants under composition is characterized.

Foundations for further study of knot composition using polynomial invariants.

Abstract

Knot theory is the Mathematical study of knots. In this paper we have studied the Composition of two knots. Knot theory belongs to Mathematical field of Topology, where the topological concepts such as topological spaces, homeomorphisms, and homology are considered. We have studied the basics of knot theory, with special focus on Composition of knots, and knot determinants using Alexander Polynomials. And we have introduced the techniques to generalize the solution of composition of knots to present how knot determinants behave when we compose two knots.