Classical link invariants from the framizations of the Iwahori-Hecke algebra and the Temperley-Lieb algebra of type $A$

TL;DR

This paper introduces new classical link invariants derived from Yokonuma-Hecke algebras and their Temperley-Lieb analogues, which are distinct from traditional invariants like the Homflypt and Jones polynomials.

Contribution

It constructs novel 2-variable and 1-variable link invariants from Yokonuma-Hecke and related algebras, expanding the landscape of link invariants beyond classical polynomials.

Findings

New 2-variable invariants from Yokonuma-Hecke algebras

New 1-variable invariants from Temperley-Lieb analogues

Establishment of algebraic isomorphisms involving braid and tie algebras

Abstract

In this paper we first present the construction of the new 2-variable classical link invariants arising from the Yokonuma-Hecke algebras ${\rm Y}_{d,n}(q)$, which are not topologically equivalent to the Homflypt polynomial. We then present the algebra ${\rm FTL}_{d,n}(q)$ which is the appropriate Temperley-Lieb analogue of ${\rm Y}_{d,n}(q)$, as well as the related 1-variable classical link invariants, which in turn are not topologically equivalent to the Jones polynomial. Finally, we present the algebra of braids and ties which is related to the Yokonuma-Hecke algebra, and also its quotient, the partition Temperley-Lieb algebra ${\rm PTL}_n(q)$ and we prove an isomorphism of this algebra with a subalgebra of ${\rm FTL}_{d,n}(q)$.