Georgia Institute of Technology College of Sciences

The Jones polynomial was first defined by Vaughan Jones as a "trace function" on an algebra discovered via operator algebras. It was discovered that the polynomial satisfies certain skein relations. The HOMFLY polynomial was discovered through both skein relations and a "lift" of the trace function on the Jones algebra to the "Hecke algebra". Another 2-variable polynomial called the Kauffman polynomial was discovered purely via skein relations. In this talk, we discuss how the process started by Jones was reversed for this polynomial. More precisely, we will show how Birman, Wenzl, and Murakami constructed the BMW algebra and a trace function that yields the Kauffman polynomial. We will discuss the significance of the Kauffman polynomial as well as some relationships between the BMW, Hecke, and Jones algebras.