SL 2 ( C ) -holonomy invariants of links Calvin McPhail-Snyder June 1, 2020 UC Berkeley
Acknowledgements ❼ I would like to thank Martin Bobb and Allison N. Miller for organizing the Nearly Carbon Neutral Geometric Topology Conference, ❼ and also to thank Carmen Caprau and Christine Ruey Shan Lee for organizing the session on quantum invariants and inviting me to speak. ❼ Much of the mathematics I will present is due to Kashaev-Reshetikhin and Blanchet, Geer, Patureau-Mirand, and Reshetikhin, although I will also discuss some of my own work (mostly in this part.) 1
Introduction
Overview Part I (previously) General idea of and motivation for a holonomy invarant of a link L with a representation π 1 ( S 3 \ L ) → G . Part II (now) Construction of a holonomy invariant for G = SL 2 ( C ) due to Blanchet, Geer, Patureau-Mirand, and Reshetikhin [Bla+20] ❼ I will also discuss my recent work [McP20] interpreting their invariant in terms of the SL 2 ( C )-twisted Reidemeister torsion. ❼ The plan is: 1. Discuss some properties of the BGPR construction and how it relates to other link invariants 2. Give an overview of the technical aspects of the construction 2
What is the BGPR invariant?
What is the BGPR invariant? BGPR invariant ❼ L a link in S 3 and ρ a representation π 1 ( S 3 \ L ) → SL 2 ( C ). ❼ Pick an integer r ≥ 2. ❼ Pick some r th roots: Let x i be a meridian of the i th component of L such that ρ ( x i ) has eigenvalues λ ± i . Choose r th roots µ r i = λ i . The r th BGPR invariant is a complex number F r ( L , ρ, { µ i } ) defined up to an overall r 2 th root of 1. Furthermore, F r is invariant under global conjugation of ρ (i.e. it is gauge invariant. ) Caveat F is currently only defined for λ i � = ± 1. A fix is in preparation. 3
Abelian case Here’s a simple case: ❼ For any link L , pick t � = 0. Then there is a representation sending every meridian x to � � t ρ ( x ) = t − 1 ❼ Any ρ with abelian image (which avoids ± 1 as eigenvalues) is conjugate to one of this type, but maybe different t i for each component. ❼ In this special case, F r ( L , ρ, { t 1 / r } ) is equal to the r th i Akutsu-Deguchi-Ohtsuki (ADO) invariant. ❼ For r = 2, F 2 ( L , ρ, {√ t i } ) is the Conway potential (Alexander polynomial, Reidemeister torsion) 4
Nonabelian case ❼ Now let ρ be a representation with nonabelian image. ❼ F r ( L , ρ, { µ i } ) is a deformation of the ADO invariant discussed previously. ❼ Idea is that the t i are now the eigenvalues λ i . ❼ The novely in the BGPR construction is that we can use nonabelian ρ . ❼ In the special case r = 2 we can say explicitly what we mean by “a deformation.” 5
Torsions of link exteriors Here’s a related abelian/nonabelian link invariant. ❼ The Reidemeister torsion of S 3 \ L is constructed using the ρ -twisted homology H ∗ ( S 3 \ L ; ρ ). ❼ For ρ sufficiently nontrivial, H ∗ ( S 3 \ L , ρ ) is acyclic and we can extract a number τ ( L , ρ ), the torsion . ❼ For abelian representations ρ ( x ) = t we get a Laurent polynomial, the Alexander polynomial. � � t ❼ For abelian representations ρ ( x ) = we get the square of t − 1 the Alexander polynomial. ❼ For nonabelian representations we get the twisted or nonabelian Reidemeister torsion. 6
BGPR invariant versus torsion Theorem [McP20] Let L be a link in S 3 and ρ : π 1 ( S 3 \ L ) a representation such that ρ ( x ) never has 1 as an eigenvalue for any meridian x of L . Then F 2 ( L , ρ, { µ i } ) F 2 ( L , ρ, { µ i } ) = τ ( L , ρ ) for any choice of roots µ i . Here L is the mirror image of L . 7
r = 2 BGPR is a nonabelian Conway potential One way to understand this theorem: ❼ If you compute the torsion using the abelian represenation � � t x �→ t − 1 it factors into two pieces because the matrix has two blocks. Each piece is equal to the Conway potential of the link. ❼ For a nonabelian representation, it is not obvious how to factor the torsion into two pieces. But this is exactly what the BGPR invariant does. ❼ Therefore we could call F 2 ( L , ρ ) a nonabelian or twisted Conway potential. 8
Significance ❼ Torsions are a useful invariant, so this indicates that holonomy invariants should be useful too. ❼ For example, twisted Alexander polynomials (which are closely related) are quite useful in knot theory. ❼ It is possible to compute the hyperbolic volume of a knot complement from an asympotic limit of hyperbolically-twisted torsions. ❼ I am hopeful that a relationship between F r and the torsions for r > 2 can be developed to take advantage of this. 9
How to construct the BGPR invariant
Quantum sl 2 Definition U q = U q ( sl 2 ) is the algebra over C ( q ) generated by K ± 1 , E , F with relations EF − FE = K − K − 1 KE = q 2 EK , KF = q − 2 FK , q − q − 1 ❼ This is a q -analogue of the universal enveloping algebra of sl 2 , with K = q H . ❼ The center of U q is generated by the quantum Casimir element Ω := ( q − q − 1 ) 2 FE + qK − q − 1 K − 1 10
Quantum sl 2 at a root of unity Set q = ξ = exp( π i / r ) a 2 r th root of 1. Facts 1. U ξ is rank r 2 over the central subalgebra Z 0 := C [ K r , K − r , E r , F r ] 2. Z 0 is a commutative Hopf algebra, so it’s the algebra of functions on a group. Specifically, Spec Z 0 ∼ = SL 2 ( C ) ∗ 3. The center of U ξ is generated by Z 0 and the Casimir Ω (subject to a polynomial relation.) We will get to the difference between SL 2 ( C ) ∗ and SL 2 ( C ) in a bit. 11
Grading on representations ❼ Closed points χ ∈ Spec Z 0 are homomorphisms χ : Z 0 → C . ❼ We associate � � κ − ǫ ∈ SL 2 ( C ) (1 − ǫφ ) κ − 1 φ ↔ ǫ φ/κ χ ( K r ) = κ, χ ( E r ) = χ ( F r ) = ( q − q − 1 ) r , ( q − q − 1 ) r ❼ A representation V with SL 2 ( C )-grading χ is one where every Z ∈ Z 0 acts by χ ( Z ). We say V has character χ . 12
Grading on representations Theorem If the the matrix associated to χ does not have ± 1 as an eigenvalue, then: ❼ Every representation with character χ is projective, irreducible, and r -dimensional. ❼ There are r isomorphism classes of these, parametrized by the action of Ω. The idea is that we associate a strand with holonomy corresponding to χ to a representation with character χ . We needed the extra data of the choice { µ i } of roots to know which of the r irreps to pick. 13
Braiding on representations ❼ There is an automorphism ˇ R : U ξ ⊗ U ξ → U ξ ⊗ U ξ satisfying the braid relations. ❼ If U ξ were quasitriangular ˇ R would be conjugation by the universal R -matrix followed by swapping the tensor factors, but for technical reasons only the outer autormorphism ˇ R exists. ❼ ˇ R acts nontrivially on Z 0 ⊗ Z 0 , so it induces a map of modules V χ 1 V χ 2 V χ 4 V χ 3 corresponding to the colored braid groupoid action on colors. Notice that the isomorphism classes of each strand change. 14
Problems with the braiding 1. The above action on modules is only defined up to a scalar; we can mostly fix this, but we get the root-of-unity indeterminacy in F r . 2. The map ( χ 1 , χ 2 ) → ( χ 4 , χ 3 ) is not the conjugation action on SL 2 ( C ), but something more complicated. Fixing 2 is harder. It is related to the fact that Spec Z 0 is really the Poisson dual group ��� � � ��� κ 0 1 ǫ SL 2 ( C ) ∗ := , ⊆ GL 2 ( C ) × GL 2 ( C ) φ 1 0 κ SL 2 ( C ) ∗ is birationally equivalent as a variety, but not isomorphic as group, to SL 2 ( C ). The equivalence is ( x + , x − ) ↔ x + ( x − ) − 1 15
Factorized biquandle ❼ The braid group action ( g 1 , g 2 ) → ( g − 1 1 g 2 g 1 , g 1 ) on colors is an example of a quandle , the conjugation quandle of SL 2 ( C ). ❼ A quandle is an algebraic structure that describes colors on arcs of knot diagrams. There are more general ones than conjugation. ❼ The more complicated action on SL 2 ( C ) ∗ colors is a generalization called a biquandle . ❼ It can be shown that the biquandle is a factorization of the conjugation quandle of SL 2 ( C ). 16
Braid groupoid representations from U ξ ❼ Instead of a representation of the colored braid groupoid B (SL 2 ( C )), we get a representation of a different, closely related groupoid B (SL 2 ( C )) ∗ . ❼ Via the theory of qunandle factorizations developed in [Bla+20], we can use closures of braids in B (SL 2 ( C )) ∗ to represent SL 2 ( C )-links. ❼ Short version: The grading on U ξ -modules is not quite right, so we have to use a nonstandard coordinate system for representations of link complements. 17
Modified traces ❼ To get representations of links (closed braids) we need a way to take traces/closures. ❼ Problem: The algebra U ξ is not semisimple and the quantum dimensions of the irreps we want to use are all zero. In particular, all our link invariants will be 0. ❼ One way to fix this: Take the partial quantum trace of F ( β ) : V g 1 ⊗ · · · V g n → V g 1 ⊗ · · · ⊗ V g n to get a map ptr( F ( β )) : V g 1 → V g 1 . (That is, write your link as a 1-1 tangle .) 18
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