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Using Cluster Algebras to Compute Homologies on Two-Strand Braid Varieties

Student-Run Research Seminar

Speaker: Tonie Scroggin, UC Davis
Location: 3106 MSB
Start time: Wed, Jan 25 2023, 12:00PM

In this introductory talk, we'll discuss the relation between braid varieties and cluster algebras. In 1923, Alexander proved that every knot/link can be represented as the closure of a braid. Similar to the Reidemeister moves on knots, any two closures represent the same link if and only if the braids are related by a sequence of stabilizations and conjugations, which are referred to as Markov moves. This allows one to construct link invariants by assigning objects to the crossings $\sigma_i^\pm$, verifying that braid relations are satisfied, defining a closure operation and checking that the result is invariant under Markov moves. We define the braid variety. The homology of a braid variety is related to the Khovanov-Rozansky homology of a corresponding link. The braid variety is isomorphic to a positroid variety; therefore, the braid variety has a cluster structure. Any cluster structure has a canonical 2-form with constant coefficients in all cluster charts, which gives an interesting class in de Rham cohomology of degree 2, and an interesting operator in link homology. Using the cluster structure, we compute the homology of the braid variety using the De Rham cohomology.

Pizza at 11:50am