New link invariants and Polynomials (I), oriented case

TL;DR

This paper introduces new oriented link invariants derived from skein relations, generalizing known polynomials like HOMFLYPT, and demonstrates their enhanced ability to distinguish complex link features such as mutants and crossing numbers.

Contribution

It constructs two new classes of link invariants with coefficients in a commutative ring, generalizing the HOMFLYPT polynomial and enhancing link distinction capabilities.

Findings

Type one invariant generalizes HOMFLYPT polynomial.

Modified invariants can distinguish mutants.

Parameterized invariants provide crossing number information.

Abstract

Given any oriented link diagram, two types of new knot invariants are constructed. They satisfy some generalized skein relations. The coefficients of each invariant is from a commutative ring. Homomorphisms and representations of those rings define new link invariants. For example, the HOMFLYPT polynomial with three variables. In this sense, type one invariant is a generalization of the HOMFLYPT polynomial. Those invariants can also be modified by writhe and parameterized to get more powerful invariants. For example, the modified type one invariant distinguishes mutants, and the parameterized invariants produces information for crossing number.