Tangles in affine Hecke algebras

Abstract

The affine Hecke algebra $\dot H_n$ of type $A$ is often presented as a quotient of the braid algebra of $n$-braids in the annulus. This leads to diagrammatic representations in terms of braids in the annulus, subject to a quadratic relation for the simple Artin braids, as in the description by Graham and Lehrer in \cite{GL03}. I show here that the use of more general framed oriented $n$-tangle diagrams in the annulus, subject to the Homfly skein relations, produces an algebra which is isomorphic to $\dot H_n$ with an extended ring of coefficients. This setting allows the use of some attractive diagrams for elements of $\dot H_n$, using closed curves as well as braids, and gives neat pictures for its central elements.