Conway type invariants of links and Kauffman's method

TL;DR

This paper introduces Conway algebras and Kauffman algebras to construct link invariants, including the Homflypt polynomial and Kauffman polynomial, providing detailed proofs and formalism for these invariants.

Contribution

It develops a formal framework for Conway and Kauffman algebras, extending the methods for constructing link invariants with detailed proofs.

Findings

Proof of the existence of Conway type invariants.

Construction of Kauffman algebra formalism.

Explicit description of Kauffman polynomial.

Abstract

In this chapter (Chapter III) we introduce the concept of Conway algebras (the notion related to entropic magmas) and describe invariants of links yielded by (partial) Conway algebras (including the Homflypt polynomial and signatures). We present, in detail, a proof (following the original Przytycki-Traczyk 1984 proof) of the existence of Conway type invariants. Then we describe Kauffman's method of constructing link invariants, in particular, giving the Kauffman polynomial of two variables. We develop a formalism, Kauffman algebras, analogous to that of Conway algebras. This chapter, along with chapters II, IV, and V of the book (already on arXiv), form the core of the first part of the book.