Related papers: Fusion Trees and Homological Representations
The ADO invariants are a sequence of non-semisimple quantum invariants coming from the representation theory of the quantum group $U_q(sl(2))$ at roots of unity. Ito showed that these invariants are sums of traces of quotients of…
We give a formula of the colored Alexander invariant in terms of the homological representation of the braid groups which we call truncated Lawrence's representation. This formula generalizes the famous Burau representation formula of the…
Recent work suggests that topological features of certain quantum gravity theories can be interpreted as particles, matching the known fermions and bosons of the first generation in the Standard Model. This is achieved by identifying…
We study homological representations of mapping class groups, including the braid groups. These arise from the twisted homology of certain configuration spaces, and come in many different flavours. Our goal is to give a unified general…
We provide a homological model for a family of quantum representations of mapping class groups arising from non-semisimple TQFTs (Topological Quantum Field Theories). Our approach gives a new geometric point of view on these…
This paper reinterprets Alexander-type invariants of knots via representation varieties of knot groups into the group $\textrm{AGL}_1(\mathbb{C})$ of affine transformations of the complex line. In particular, we prove that the coordinate…
Finite Temperley-Lieb (TL) algebras are diagram-algebra quotients of (the group algebra of) the famous Artin's braid group $B_N$, while the affine TL algebras arise as diagram algebras from a generalized version of the braid group. We study…
We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket…
For groups of a topological origin, such as braid groups and mapping class groups, an important source of interesting and highly non-trivial representations is given by their actions on the twisted homology of associated spaces; these are…
Knot theory is an active field of mathematics, in which combinatorial and computational methods play an important role. One side of computational knot theory, that has gained interest in recent years, both for complexity analysis and…
We study the density of the Burau representation from the perspective of a non-semisimple TQFT at a fourth root of unity. This gives a TQFT construction of Squier's Hermitian form on the Burau representation with possibly mixed signature.…
Coloured Alexander polynomials form a sequence of non-semisimple quantum invariants coming from the representation theory of the quantum group $U_q(sl(2))$ at roots of unity. This sequence recovers the original Alexander polynomial as the…
Topological Quantum Field Theories (TQFTs) pertinent to some emergent low energy phenomena of condensed matter lattice models in 2+1 and 3+1D are explored. Many of our field theories are highly-interacting without free quadratic analogs.…
We study the kernels of representations of mapping class groups of surfaces on twisted homologies of configuration spaces. We relate them with the kernel of a natural twisted intersection pairing: if the latter kernel is trivial then the…
We review the q-deformed spin network approact to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. These methods produce a concise proof…
We propose a new non-commutative generalization of the representation variety and the character variety of a knot group. Our strategy is to reformulate the construction of the algebra of functions on the space of representations in terms of…
The spin network quantum simulator relies on the su(2) representation ring (or its q-deformed counterpart at q= root of unity) and its basic features naturally include (multipartite) entanglement and braiding. In particular, q-deformed spin…
We show that any of the new knot invariants obtained from Chern-Simons theory based on an arbitrary non-abelian gauge group do not distinguish isotopically inequivalent mutant knots and links. In an attempt to distinguish these knots and…
For non-abelian simple objects in a unitary modular category, the density of their braid group representations, the #P-hard evaluation of their associated link invariants, and the BQP-completeness of their anyonic quantum computing models…
Various properties of a class of braid matrices, presented before, are studied considering $N^2 \times N^2 (N=3,4,...)$ vector representations for two subclasses. For $q=1$ the matrices are nontrivial. Triangularity $(\hat R^2 =I)$…