arXiv subjects

Konstantinos Karvounis

Publications and source records attributed to Konstantinos Karvounis.

4 recordsLinked to original sources

arXiv2019

Braid group action on projective quantum $\mathfrak{sl}(2)$ modules

We define a family of the braid group representations via the action of the $R$-matrix (of the quasitriangular extension) of the restricted quantum $\mathfrak{sl}(2)$ on a tensor power of a simple projective module. This family is an extension of the Lawrence representation specialized at roots of unity. Although the center of the braid group has finite order on the specialized Laurence representations, this action is faithful for our extension.

math.GT↗

arXiv2019

The BMM symmetrising trace conjecture for groups $G_4,\,G_5,\,G_6,\,G_7,\,G_8$

We prove the BMM symmetrising trace conjecture for the exceptional complex reflection groups $G_4,\,G_5,\,G_6,\,G_7,\,G_8$ using a combination of algorithms programmed in different languages (C++, SAGE, GAP3, Mathematica). Our proof depends on the choice of a suitable basis for the generic Hecke algebra associated with each group.

math.RT↗

arXiv2016

Identifying the invariants for classical knots and links from the Yokonuma-Hecke algebras

In this paper we announce the existence of a family of new $2$-variable polynomial invariants for oriented classical links defined via a Markov trace on the Yokonuma-Hecke algebra of type $A$. Yokonuma-Hecke algebras are generalizations of Iwahori-Hecke algebras, and this family contains the Homflypt polynomial, the famous $2$-variable invariant for classical links arising from the Iwahori-Hecke algebra of type $A$. We show that these invariants are topologically equivalent to the Homflypt polynomial on knots, but not on links, by providing pairs of Homflypt-equivalent links that are distinguished by our invariants. In order to do this, we prove that our invariants can be defined diagrammatically via a special skein relation involving only crossings between different components. We further generalize this family of invariants to a new $3$-variable skein link invariant which is stronger than the Homflypt polynomial. Finally, we present a closed formula for this invariant, by W.B.R. Lickorish, which uses Homflypt polynomials of sublinks and linking numbers of a given oriented link.

math.GT↗

arXiv2015

On the link invariants from the Yokonuma-Hecke algebras

In this paper we study properties of the Markov trace ${\rm tr}_d$ and the specialized trace ${\rm tr}_{d,D}$ on the Yokonuma-Hecke algebras, such as behaviour under inversion of a word, connected sums and mirror imaging. We then define invariants for framed, classical and singular links through the trace ${\rm tr}_{d,D}$ and also invariants for transverse links through the trace ${\rm tr}_d$. In order to compare the invariants for classical links with the Homflypt polynomial we develop computer programs and we evaluate them on several Homflypt-equivalent pairs of knots and links. Our computations lead to the result that these invariants are topologically equivalent to the Homflypt polynomial on knots. However, they do not demonstrate the same behaviour on links.

math.GT↗