TY - GEN
T1 - A survey on temperley–lieb-type quotients from the yokonuma–hecke algebras
AU - Goundaroulis, Dimos
N1 - Publisher Copyright:
© Springer International Publishing AG 2017.
PY - 2017
Y1 - 2017
N2 - In this survey we collect all results regarding the construction of the Framization of the Temperley–Lieb algebra of type A as a quotient algebra of the Yokonuma–Hecke algebra of type A. More precisely, we present all three possible quotient algebras the emerged during this construction and we discuss their dimension, linear bases, representation theory and the necessary and sufficient conditions for the unique Markov trace of the Yokonuma–Hecke algebra to factor through to each one of them. Further, we present the link invariants that are derived from each quotient algebra and we point out which quotient algebra provides the most natural definition for a framization of the Temperley–Lieb algebra. From the Framization of the Temperley–Lieb algebra we obtain new one-variable invariants for oriented classical links that, when compared to the Jones polynomial, they are not topologically equivalent since they distinguish more pairs of non isotopic oriented links. Finally, we discuss the generalization of the newly obtained invariants to a new two-variable invariant for oriented classical links that is stronger than the Jones polynomial.
AB - In this survey we collect all results regarding the construction of the Framization of the Temperley–Lieb algebra of type A as a quotient algebra of the Yokonuma–Hecke algebra of type A. More precisely, we present all three possible quotient algebras the emerged during this construction and we discuss their dimension, linear bases, representation theory and the necessary and sufficient conditions for the unique Markov trace of the Yokonuma–Hecke algebra to factor through to each one of them. Further, we present the link invariants that are derived from each quotient algebra and we point out which quotient algebra provides the most natural definition for a framization of the Temperley–Lieb algebra. From the Framization of the Temperley–Lieb algebra we obtain new one-variable invariants for oriented classical links that, when compared to the Jones polynomial, they are not topologically equivalent since they distinguish more pairs of non isotopic oriented links. Finally, we discuss the generalization of the newly obtained invariants to a new two-variable invariant for oriented classical links that is stronger than the Jones polynomial.
UR - https://www.scopus.com/pages/publications/85041290303
UR - https://www.scopus.com/pages/publications/85041290303#tab=citedBy
U2 - 10.1007/978-3-319-68103-0_2
DO - 10.1007/978-3-319-68103-0_2
M3 - Conference contribution
AN - SCOPUS:85041290303
SN - 9783319681023
T3 - Springer Proceedings in Mathematics and Statistics
SP - 37
EP - 55
BT - Algebraic Modeling of Topological and Computational Structures and Applications
A2 - Theodorou, Doros
A2 - Kauffman, Louis H.
A2 - Lambropoulou, Sofia
A2 - Stefaneas, Petros
PB - Springer New York LLC
T2 - THALES Workshop on Algebraic Modeling of Topological and Computational Structures and Applications, AlModTopCom 2015
Y2 - 1 July 2015 through 3 July 2015
ER -