UCSB QT Seminar

Title: Skein-theoretic methods for unitary fusion categories

Abstract: Given a fusion rule $q\otimes q \cong 1\oplus\bigoplus{i=1}^{k} x_{i}$ in a unitary fusion category C, we extract information using skein-theoretic methods and a rotation operator. For instance, one can classify all associated framed link invariants when k = 1, 2 and C is ribbon. We consider an action of the rotation operator on a “canonical basis”. Assuming self-duality of the summands x_i , we will explore some properties of certain 6j-symbols using skein theory. We also explore some features of quantum invariants coming from q antisymmetrically self-dual. This is joint work with Sachin J. Valera.

Title: Moduli of principal bundles for 2-groups.

(Joint work with Eric Berry, Dan Berwick-Evans, Emily Cliff, Apurva Nakade, and Emma Phillips) 

Abstract: In this talk, we will introduce the notion of 2-groups, a categorified version of groups, and principal bundles for 2-groups. Motivated by work by Schommer-Pries towards finding a finite-dimensional model of the String group using higher categories, we will consider 2-groups arising as central extensions of a finite (ordinary) group G. Restricting to this special case, we obtain a natural construction of a G-equivariant line bundle over the moduli of principal G-bundles on a surface X, using principal 2-group bundles. This categorical construction turns out to be equivalent to the Freed-Quinn line bundle constructed using ‘transgression’, and we will briefly sketch that relationship.

Title: Noncommutative tensor triangular geometry and cohomological support varieties.

Abstract: Recently, there has been significant interest in the tensor product property for cohomological support varieties of Hopf algebras and tensor categories. We will describe a method for approaching the tensor product property by way of a noncommutative version of Balmer’s tensor triangular geometry in the general setting of a monoidal triangulated category. We prove related properties about the collections of thick one-sided and two-sided ideals of the category, and then are often able to use the universal properties of the Balmer support to obtain applications to cohomological supports. Examples arising from the representation theory of Hopf algebras will be discussed throughout. This is a joint project with Daniel Nakano and Milen Yakimov. 

Title:  Generators and Relations for Rep(Sp(2n)) 

Abstract: A useful technique in representation theory is to recover the category of all representations from a combinatorial subcategory. For example, Fund(Sp(2n)), the full subcategory of Rep(Sp(2n)) monoidally generated by the (finite dimensional) representations with fundamental highest weight, has idempotent completion Rep(Sp(2n)). 

In joint work with Elias, Rose, and Tatham, we define a category by generators and relations and argue this category maps to Fund(Sp(2n)). Then by combining skein theoretic arguments with a combinatorial result due to Sundaram we deduce the functor is an equivalence of monoidal categories. This solves the type C_n case of a problem Kuperberg posed in his 1996 paper on rank two spiders, and by passing to the Karoubi envelope describes a universal property for the category Rep(Sp(2n)).

In the talk I will state the results, then try to give an idea of how some of the arguments work by illustrating them in the case of Sp(4) and Sp(6). 

Title:  Classical and quantum traces coming from SL_n(C) and U_q(sl_n)

Abstract:  Let S be a punctured surface.  The SL_2-skein algebra of S is a non-commutative algebra, whose elements are represented by knots K in the thickened surface S x [0,1] subject to certain relations.  The skein algebra is a quantum deformation of the SL_2(C)-character variety of S, depending on a deformation parameter q.  Bonahon and Wong constructed an injective algebra map--called the quantum trace map--from the skein algebra of S into a simpler non-commutative algebra, the latter of which can be thought of as a quantum Teichmüller space of S.  This mapping associates to a knot K in S x [0,1] a Laurent polynomial in q-commuting variables X_i, recovering in the classical case q=1 the polynomial expressing the traces of monodromies of hyperbolic structures on S when written in terms of Thurston's shear-bend coordinates for Teichmüller space.  We discuss a SL_n-version of this invariant naturally coming from Fock and Goncharov’s classical and quantum higher Teichmüller theory.  

Title:  Stated Skein Modules of Marked 3-Manifolds 

Abstract: In this talk we will explore joint work with Thang Le on the stated skein modules of marked 3-manifolds.  These generalize ordinary skein modules by allowing for not only embedded links, but also stated tangles which meet boundary markings.  Our main goal will be to define and look at the properties of the Chebyshev-Frobenius homomorphism of stated skein modules specialized at roots of unity.  We will finish off by discussing the role of the Chebyshev-Frobenius homomorphism in some conceptual frameworks for stated skein modules.  There will be many pictures. 

Title: Guts, Volume, and Skein Modules of 3-Manifolds

Abstract: When looking at hyperbolic alternating knots in S^3, there is a relation between the twist number, the Jones polynomial, and the volume of the knot complement. Little is known for general hyperbolic links, or links in other manifolds. We consider hyperbolic links that admit alternating projections on surfaces in compact, irreducible 3-manifolds. Under some mild hypotheses, we are able to show that volume of the link complement is bounded below in terms of a Kauffman bracket function defined on link diagrams on the surface. Further, if the manifold is a thickened surface, we can construct a Jones-type polynomial that is an isotopy invariant that leads to a 2-sided linear bound on the volume of hyperbolic alternating links in the thickened surface. 

Title: Understanding the structure of the framing skein module and the Kauffman bracket skein module

Abstract: Skein modules were introduced in 1987 by Jozef H. Przytycki as generalisations of the various polynomial link invariants in the 3-sphere to arbitrary 3-manifolds. Over time they have evolved into one of the most important objects in knot theory and quantum topology having strong ties with algebraic and hyperbolic geometry, quantum cluster algebras, and the Witten-Reshetikhin-Turaev 3-manifold invariants, to name a few. In this talk we will focus on the structure of two different skein modules: the framing skein module and the Kauffman bracket skein module of 3-manifolds: 

1. We show that the only way of changing the framing of a link by ambient isotopy in an oriented 3-manifold is when the manifold admits a properly embedded non-separating 2-sphere. This change of framing is given by the Dirac trick. We use this result to completely determine the structure of the framing skein module and show that it detects the presence of non-separating 2-spheres by way of torsion. 

2. We disprove a twenty-two-year-old theorem about the structure of the Kauffman bracket skein module of the connected sum of two handlebodies. We achieve this by analysing handle slidings on compressing discs in a handlebody.