Constructing Hecke-type Structures, their Representations and Applications

Abstract

This thesis is primarily concerned with the construction of a large Hecke-type structure called the double ane Q-dependent braid group. The signicance of this structure is that it is located at the top level of the hierarchy of all other structures that are known to be related to the braid group. In particular, as specialisations we obtain the Hecke algebra, in addition to the ane Hecke algebra, even the double ane Hecke algebra and also the elliptic braid group. To render the algebraic description of this group more accessible, we present an intuitive graphical representation that we have specically developed to fully capture all of its structure. Contained within this representation are representations of all of the afore mentioned algebras which all contain the braid group as primary element. We also present nite dimensional matrix representations of ane Hecke algebras, emerging from tangles. Using these tangles we also obtain representations of the Temperley-Lieb algebra and the ane braid group. We conclude this thesis with our interpretation of the central role of the Hecke algebra in the development of knot theory. More specically we explicitly derive the HOMFLY and Jones polynomials