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# Mathematical Research Letters

## Volume 24 (2017)

### Number 2

### Framization of the Temperley–Lieb algebra

Pages: 299 – 345

DOI: https://dx.doi.org/10.4310/MRL.2017.v24.n2.a3

#### Authors

#### Abstract

We propose a framization of the Temperley–Lieb algebra. The framization is a procedure that can briefly be described as the adding of framing to a known knot algebra in a way that is both algebraically consistent and topologically meaningful. Our framization of the Temperley–Lieb algebra is defined as a quotient of the Yokonuma–Hecke algebra. The main theorem provides necessary and sufficient conditions for the Markov trace defined on the Yokonuma–Hecke algebra to pass through to the quotient algebra. Using this we construct $1$-variable invariants for classical knots and links, which, as we show, are not topologically equivalent to the Jones polynomial.

#### Keywords

Temperley–Lieb algebra, Yokonuma–Hecke algebra, Markov trace, link invariants

#### 2010 Mathematics Subject Classification

20C08, 20F36, 57M25

This research has been co-financed by the European Union (European Social Fund - ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) - Research Funding Program: THALES: Reinforcement of the interdisciplinary and/or inter-institutional research and innovation.

Received 28 January 2015

Accepted 12 February 2016

Published 24 July 2017