arXiv subjects
Publications and source records attributed to Wade Bloomquist.
We extend the notion of congruence subgroups of the braid group to the virtual braid group using an extension of the integral Burau representation. We prove that the level 2 congruence subgroup of the virtual braid group is the pure virtual braid group, recovering a virtual analogue of a result of Arnol'd. We pose several questions which highlight the difference between the classical and virtual braid groups.
We introduce a joint generalization, called LRY skein algebras, of Kauffman bracket skein algebras (of surfaces) that encompasses both Roger-Yang skein algebras and stated skein algebras. We will show that, over an arbitrary ground ring which is a commutative domain, the LRY skein algebras are domains and have degenerations (by filtrations) equal to monomial subalgebras of quantum tori. For surfaces without interior punctures, this integrality generalizes a result of Moon and Wong to the most general ground ring. We also calculate the Gelfand-Kirillov dimension of LRY algebras and show they are Noetherian if the ground ring is. Moreover they are orderly finitely generated. To study the LRY algebras and prove the above-mentioned results, we construct quantum traces, both the so-called X-version for all surfaces and also an A-version for a smaller class of surfaces. We also introduce a modified version of Dehn-Thurston coordinates for curves which are more suitable for the study of skein algebras as they pick up the highest degree terms of products in certain natural filtrations.
The congruence subgroups of braid groups arise from a congruence condition on the integral Burau representation $B_n \to \operatorname{GL}_{n}(\mathbb Z)$. We find the image of such congruence subgroups in $\operatorname{GL}_{n}(\mathbb Z)$-an open problem posed by Dan Margalit. Additionally, we characterize the quotients of braid groups by their congruence subgroups in terms of symplectic congruence subgroups.
The integral Burau representation provides a map from the braid group into a group of integral matrices. This allows for a definition of congruence subgroups of the braid group as the preimage of the usual principal congruence subgroups of integral matrices. We explore the structure these congruence subgroups by examining some of the quotients that may arise in the series induced by divisibility of levels. We build on the work of Stylianakis on symmetric quotients of congruence subgroups, which itself generalizes the quotient of the braid group by the pure braid group. We accomplish this by utilizing results of Newman on integral matrices and explicitly finding elements in the preimage of any transposition. Our generalization is made possible by avoiding the use of a generating set for congruence subgroups. We find further generalizations based on results of Brendle and Margalit as well as Kordek and Margalit on the level four congruence subgroup. This gives families of quotients which are not isomorphic to symmetric groups.
We study the stated skein modules of marked 3-manifolds. We generalize the splitting homomorphism for stated skein algebras of surfaces to a splitting homomorphism for stated skein modules of 3-manifolds. We show that there exists a Chebyshev-Frobenius homomorphism for the stated skein modules of 3-manifolds which extends the Chebyshev homomorphism of the skein algebras of unmarked surfaces originally constructed by Bonahon and Wong. Additionally, we show that the Chebyshev-Frobenius map commutes with the splitting homomorphism. This is then used to show that in the case of the stated skein algebra of a surface, the Chebyshev-Frobenius map is the unique extension of the dual Frobenius map (in the sense of Lusztig) of $\mathcal{O}_{q^2}(SL(2))$ through the triangular decomposition afforded by an ideal triangulation of the surface. In particular, this gives a skein theoretic construction of the Hopf dual of Lusztig's Frobenius homomorphism. A second conceptual framework is given, which shows that the Chebyshev-Frobenius homomorphism for the stated skein algebra of a surface is the unique restriction of the Frobenius homomorphism of quantum tori through the quantum trace map.
We propose an encoding for topological quantum computation utilizing quantum representations of mapping class groups. Leakage into a non-computational subspace seems to be unavoidable for universality in general. We are interested in the possible gate sets which can emerge in this setting. As a first step, we prove that for abelian anyons, all gates from these mapping class group representations are normalizer gates. Results of Van den Nest then allow us to conclude that for abelian anyons this quantum computing scheme can be simulated efficiently on a classical computer. With an eye toward more general anyon models we additionally show that for Fibonnaci anyons, quantum representations of mapping class groups give rise to gates which are not generalized Clifford gates.
We prove asymptotic faithfulness for the quantum $\mathrm{Sp}(4)$ mapping class group representation. This provides the first example of asymptotic faithfulness lying outside of the $A_n$ family. The methods used are generalized from the proof of asymptotic faithfulness for skein $SU(2)_k$ mapping class group representations. In short, for any noncentral mapping class a comparison vector is found which allows for the mapping class to be detected.
A tensor category is multiplicity-free if for any objects $A,B,C$ we have that $\mathrm{Hom}(A\otimes B\otimes C,\mathbb{C})$ is either $0$ or $1$ dimensional. It is known that $Rep^{uni}(U_q(\mathfrak{sp}(4)))$ is not multiplicty-free. We find a full subcategory of $Rep^{uni}(U_q(\mathfrak{sp}(4)))$ which is multiplicty-free. A description of the dimension of these $\mathrm{Hom}$ spaces is given for this subcategory, including when $q$ is a root of unity. The methods used arise from the description, given by Kuperberg, of $Rep^{uni}(U_q(\mathfrak{sp}(4)))$ as a spider. The main tool is the recursive definition of clasps given by Kim. In particular, we provide an appropriate notion of admissibility when looking at the $\mathrm{Sp}(4)_k$ ribbon graph invariants with restricted edge labels.
We generalize the asymptotic faithfulness of the skein quantum $SU(2)$ representations of mapping class groups of orientable closed surfaces to skein $SU(3)$. Skein quantum representations of mapping class groups are different from the Reshetikin-Turaev ones from quantum groups or geometric quantization because they are given by different modular tensor categories. We conjecture asymptotic faithfulness holds for skein quantum $G$ representations when $G$ is a simply-connected simple Lie group. The difficulty for such a generalization lies in the lack of an explicit description of the fusion spaces with multiplicities to define an appropriate complexity of state vectors.