Appears in collection : 2026 - T1 - WS2 - Bridging visualization and understanding in Geometry and Topology

The Temperley-Lieb algebra first arose as a matrix algebra describing transfer functions in statistical mechanics models such as the Potts and Ising models. The algebra acquired a formal definition in terms of generators and relations that allowed its representations to be identified in multiple contexts. In the early 1980's Vaughan Jones found the algebra once again in a context between mathematics and physics as an algebra of projectors that arose in a tower construction of von Neumann algebras. For this context, Jones investigated the formally defined algebra and its matrix representations and he constructed a trace function on the Temperley-Lieb (TL) algebra (a function $tr$ to a commutative ring such that $tr(ab) = tr(ba)$ for $ab$ a product in the (non-commutative) Temperley-Lieb algebra) and he discovered a representation of the Artin Braid group to the TL algebra. By composing this representation with the trace $tr$, Jones defined an invariant of braids that could be modified via the Markov Theorem for braids, knots and links to produce a polynomial invariant of knots that is now known as the Jones polynomial. The speaker discovered knot diagrammatic and combinatorial interpretations of the Jones polynomial and the Temperley-Lieb algebra that allow the polynomial to be seen as part of a generalized Potts model partition function defined on planar link diagrams and planar graphs. The combinatorial interpretation of the Temperley-Lieb algebra allows the Jones trace to be interpreted as a loop count for closures of Connection Monoid representations of the Temperley-Lieb algebra. The multiplicative structure of the Temperley-Lieb algebra is represented in the speaker's work by a Connection Monoid whose elements are families of planar connections between two rows of points where the connections can go from row to row or from one row to the other.

The talk will begin with the formal definition of the TL monoid and will show how it is modeled by the Connection Monoid and similarly with the TL Category and a Connection Category. This interpretation allows us to see answers to algebra questions about the Temperley-Lieb Monoid that would be invisible without the combinatorial interpretation. In particular we will show how the structure of repeated powers of elements in TL appears and how idempotents correspond to generalized meanders. A meander is a Jordan curve in the plane cut through transversely by a straight line. The fascinating and highly visual combinatorics of the meanders informs the structure of the TL algebra via the way meanders correspond to factorizations of the identity in the Temperley-Lieb Category.