The elements of the Temperley-Lieb Algebra, TLn, can be realized as crossingless connections between an even number of marked boundary points in a rectangle, and multiplication in TLn is given by concatenation of these crossingless connections. This talk will begin with exposition on the Temperley-Lieb algebra—especially in its connections with knot theory—before describing a bilinear form on the Temperley-Lieb algebra defined by “counting unknots”. We will generalize this bilinear form and its corresponding Gram determinant to a set of crossingless connections in the Mobius band.
