A 3-Variable Bracket

TL;DR

This paper introduces a modified knot invariant based on the Kauffman bracket that is algebraically stronger and can distinguish some links that the original bracket cannot, using a new algebraic framework.

Contribution

It presents a new algebraic invariant derived from the Kauffman bracket, living in a quotient ring, which is stronger than the traditional bracket for distinguishing links.

Findings

The new invariant is algebraically stronger than the traditional bracket.

Any link distinguished by the usual bracket is also distinguished by the new invariant.

An open problem remains to find knots distinguished by the new invariant but not by the traditional bracket.

Abstract

Kauffman's bracket is an invariant of regular isotopy of knots and links which since its discovery in 1985 it has been used in many different directions: (a) it implies an easy proof of the invariance of (in fact, it is equivalent to) the Jones polynomial; (b) it is the basic ingredient in a completely combinatorial construction for quantum 3-manifold invariants; (c) by its fundamental character it plays an important role in some theories in Physics; it has been used in the context of virtual links; it has connections with many objects other objects in Mathematics and Physics. I show in this note that, surprisingly enough, the same idea that produces the bracket can be slightly modified to produce algebraically stronger regular isotopy and ambient isotopy invariants living in the quotient ring $R/I$, where the ring $R$ and the ideal $I$ are: \begin{center} $R=\Z[\alpha,\beta,\delta]$, $I=< p_1, p_2 >$, with $p_1=\alpha^2 \delta + 2 \alpha \beta \delta ^2 -\delta ^2+\beta ^2 \delta, p_2=\alpha \beta \delta ^3+\alpha ^2 \delta ^2+\beta ^2 \delta ^2+\alpha \beta \delta -\delta.$ \end{center} It is easy to prove that any pair of links distinguished by the usual bracket is also distinguishable by the new invariant. The contrary is not necessarily true. However, a explicit example of a pair of knots not distinguished by the bracket and distinguished by this new invariant is an open problem.