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In this paper we introduce distinct approaches to loop braid groups, a generalisation of braid groups, and unify all the definitions that have appeared so far in literature, with a complete proof of the equivalence of these definitions.…
Any representation of data involves arbitrary investigator choices. Because those choices are external to the data-generating process, each choice leads to an exact symmetry, corresponding to the group of transformations that takes one…
A virtual link diagram is called normal if the associated abstract link diagram is checkerboard colorable, and a virtual link is normal if it has a normal diagram as a representative.In this paper, we introduce a method of converting a…
Current work in lexical distributed representations maps each word to a point vector in low-dimensional space. Mapping instead to a density provides many interesting advantages, including better capturing uncertainty about a representation…
We review some recent advances in modular representation theory of symmetric groups and related Hecke algebras. We discuss connections with Khovanov-Lauda-Rouquier algebras and gradings on the blocks of the group algebras $F\Sigma_n$, which…
We relativise the Thomassen--Woess definition of accessibility in graphs, defining what it means for a graph to be accessible relative to a peripheral system. In the case of locally finite, quasi-transitive graphs, we characterise relative…
In this paper, we use skein-theoretic techniques to classify all virtual knot polynomials and trivalent graph invariants with certain smallness conditions. The first half of the paper classifies all virtual knot polynomials giving…
Every link diagram can be represented as a signed ribbon graph. However, different link diagrams can be represented by the same ribbon graphs. We determine how checkerboard colourable diagrams of links in real projective space, and virtual…
We introduce a recoupling theory for virtual braided trees. This recoupling theory can be utilized to incorporate swap gates into anyonic models of quantum computation.
We define virtual braid groups of type B and construct a morphism from such a group to the group of isomorphism classes of some invertible complexes of bimodules up to homotopy.
The graph braid group of a complete bipartite graph is the fundamental group of a configuration space of points on the graph, which is a CAT(0) cube complex. We combine an analysis of the topology of links of vertices in this complex, the…
A diagram of groupoid correspondences is a homomorphism to the bicategory of \'etale groupoid correspondences. We study examples of such diagrams, including complexes of groups and self-similar higher-rank graphs. We encode the diagram in a…
We introduce the notion of stable representations, -- it is a new class of the representations of a certain class of groups which defined with positive definite functions which generalize the classical notion of the characters (or trace).…
We introduce the universal virtual braid group $UV_n(c)$, which provides a unified algebraic framework for virtual braid--type structures with $c$ types of crossings and admits natural quotient maps onto the standard families in the…
There is a classical connection between the representation theory of the symmetric group and the general linear group called Schur-Weyl duality. Variations on this principle yield analogous connections between the symmetric group and other…
Interactions and relations between objects may be pairwise or higher-order in nature, and so network-valued data are ubiquitous in the real world. The "space of networks", however, has a complex structure that cannot be adequately described…
We investigate the braid group representations arising from categories of representations of twisted quantum doubles of finite groups. For these categories, we show that the resulting braid group representations always factor through finite…
Quantum walks provide a natural framework to approach graph problems with quantum computers, exhibiting speedups over their classical counterparts for tasks such as the search for marked nodes or the prediction of missing links.…
We define a notion of quantum automorphism group of Graph C*-algebras for finite, connected graphs. Under the assumption that the underlying graph does not have any multiple edge or loop, the quantum automorphism group of underlying…
Representation learning on graphs has been gaining attention due to its wide applicability in predicting missing links, and classifying and recommending nodes. Most embedding methods aim to preserve certain properties of the original graph…