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Session S10 - Categorification, Higher Representation Theory, and Homological Knot Invariants

Tuesday, July 20, 20:20 ~ 20:55 UTC-3

The Kauffman/BMW skein algebra of the torus

Peter Samuelson

University of California, Riverside, United States   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak34c487beded1daa179090896e1e2ff08').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy34c487beded1daa179090896e1e2ff08 = 'ps&#97;m&#117;&#101;ls' + '&#64;'; addy34c487beded1daa179090896e1e2ff08 = addy34c487beded1daa179090896e1e2ff08 + '&#117;cr' + '&#46;' + '&#101;d&#117;'; var addy_text34c487beded1daa179090896e1e2ff08 = 'ps&#97;m&#117;&#101;ls' + '&#64;' + '&#117;cr' + '&#46;' + '&#101;d&#117;';document.getElementById('cloak34c487beded1daa179090896e1e2ff08').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy34c487beded1daa179090896e1e2ff08 + '\'>'+addy_text34c487beded1daa179090896e1e2ff08+'<\/a>';

The BMW algebra surjects onto centralizer algebras of tensor powers of the defining representation of the type BCD quantum groups (like the Hecke algebra in type A). Kauffman found that skein relations defining this algebra can be used to define invariants of knots in $\mathbb{R}^3$. These skein relations associate algebras to surfaces, and we give a presentation of the algebra associated to the torus. At the end we ask whether there is a $q,t$ deformation of this algebra which can be used to describe BCD knot homology of (iterated) torus knots, similar to the Hall algebra of elliptic curves defined by Burban and Schiffmann.

Joint work with Hugh Morton (University of Liverpool) and Alex Pokorny (University of California, Riverside).