Related papers: Unitary Braid Matrices: Bridge between Topological…
We consider dual unitary operators and their multi-leg generalizations that have appeared at various places in the literature. These objects can be related to multi-party quantum states with special entanglement patterns: the sites are…
Braiding is a geometric concept that manifests itself in a variety of scientific contexts from biology to physics, and has been employed to classify bulk band topology in topological materials. Topological edge states can also form braiding…
Rooted in group field theory and matrix models, random tensor models are a recent background-invariant approach to quantum gravity in arbitrary dimensions. Colored tensor models (CTM) generate random triangulated orientable…
A method for compiling quantum algorithms into specific braiding patterns for non-Abelian quasiparticles described by the so-called Fibonacci anyon model is developed. The method is based on the observation that a universal set of quantum…
This is the first paper in a series where we generalize the Categorical Quantum Mechanics program (due to Abramsky, Coecke, et al) to braided systems. In our view a uniform description of quantum information for braided systems has not yet…
Motivated by topology, we develop a general theory of traces and shadows for an endobicategory, which is a~pair: bicategory $\mathbf{C}$ and endobifunctor $\Sigma\colon \mathbf C \to\mathbf C$. For a graded linear bicategory and a fixed…
We investigate the relationship between the algebra of tensor categories and the topology of framed 3-manifolds. On the one hand, tensor categories with certain algebraic properties determine topological invariants. We prove that fusion…
We study the interplay between braid group theory and topological dynamics in three dimensions. While classical braid theory has been extensively applied to surface homeomorphisms to analyze fixed and periodic points, an analogous framework…
We conjecture the existence of four independent gradings in the colored HOMFLY homology. We describe these gradings explicitly for the rectangular colored homology of torus knots and make qualitative predictions of various interesting…
While there is a general consensus about the structure of one qubit operations in topological quantum computer, two qubits are as usual a more difficult and complex story of different attempts with varying approaches, problems and…
Over-then-Under (OU) tangles are oriented tangles whose strands travel through all of their over crossings before any under crossings. In this paper we discuss the idea of gliding: an algorithm by which any tangle diagram could be brought…
Configurations of two or more branes wrapping different homology cycles of space-time are considered and the amount of supersymmetry preserved is analysed, generalising the analysis of multiple branes in flat space. For K3…
Global invertible symmetries act unitarily on local observables or states of a quantum system. In this note, we aim to generalise this statement to non-invertible symmetries by considering unitary actions of higher fusion category…
Finite-order invariants of knots in arbitrary 3-manifolds (including non-orientable ones) are constructed and studied by methods of the topology of discriminant sets. Obstructions to the integrability of admissible weight systems to…
We classify Nichols algebras of irreducible Yetter-Drinfeld modules over groups such that the underlying rack is braided and the homogeneous component of degree three of the Nichols algebra satisfies a given inequality. This assumption…
We present a sequence of diagrams of the unknot for which the minimum number of Reidemeister moves required to pass to the trivial diagram is quadratic with respect to the number of crossings. These bounds apply both in $S^2$ and in $\R^2$.
A new method for deriving universal \v{R} matrices from braid group representation is discussed. In this case, universal \v{R} operators can be defined and expressed in terms of products of braid group generators. The advantage of this…
In spacetime dimensions of 4 (i.e., 3+1) and higher, topological orders exhibit spatially extended excitations like loops and membranes, which support diverse topological data characterizing braiding, fusion, and shrinking processes,…
Braiding has attracted significant attention in physics because of its important role in describing the fundamental exchange of particles. Infusing the braiding with topological protection will make it robust against imperfections and…
In this paper a classification of Reidemeister moves, which is the most refined, is introduced. In particular, this classification distinguishes some $\Omega_3$-moves that only differ in how the three strands that are involved in the move…