Related papers: Khovanov homology of tangles: algorithm and comput…
This paper discusses relationships between topological entanglement and quantum entanglement. Specifically, we propose that for this comparison it is fundamental to view topological entanglements such as braids as "entanglement operators"…
The art of tying knots is exploited in nature and occurs in multiple applications ranging from being an essential part of scouting programs to engineering molecular knots. Biomolecular knots, such as knotted proteins, bear various cellular…
It is shown that every knot or link is the set of complex tangents of a 3-sphere smoothly embedded in the three-dimensional complex space. We show in fact that a one-dimensional submanifold of a closed orientable 3-manifold can be realised…
We review quantum field theory approach to the knot theory. Using holomorphic gauge we obtain the Kontsevich integral. It is explained how to calculate Vassiliev invariants and coefficients in Kontsevich integral in a combinatorial way…
Error-correcting codes for quantum computing are crucial to address the fundamental problem of communication in the presence of noise and imperfections. Audoux used Khovanov homology to define families of quantum error-correcting codes with…
Strongly-coupled Quantum Field Theories (QFTs) are ubiquitous in high energy physics and many-body physics, yet our ability to do precise computations in such systems remains limited. Hamiltonian Truncation is a method for doing…
This survey explores knot polynomials and their categorification, culminating in the homological invariants of knots. We begin with an overview of classical knot polynomials, progressing towards the superpolynomial and its role in unifying…
Networks based on entangled quantum systems enable interesting applications in quantum information processing and the understanding of the resulting quantum correlations is essential for advancing the technology. We show that the theory of…
We revisit Rozansky's construction of Khovanov homology for links in $S^2\times S^1$, extending it to define Khovanov homology $Kh(L)$ for links $L$ in $M^r=#^r(S^2\times S^1)$ for any $r$. The graded Euler characteristic of $Kh(L)$ can be…
We construct a spectral sequence relating the Khovanov homology of a strongly invertible knot to the annular Khovanov homologies of the two natural quotient knots. Using this spectral sequence, we re-prove that Khovanov homology…
Structure in quantum entanglement entropy is often leveraged to focus on a small corner of the exponentially large Hilbert space and efficiently parameterize the problem of finding ground states. A typical example is the use of matrix…
Topological orders are a prominent paradigm for describing quantum many-body systems without symmetry-breaking orders. We present a topological quantum field theoretical (TQFT) study on topological orders in five-dimensional spacetime…
In this paper we will present some ideas to use 3D topology for quantum computing extending ideas from a previous paper. Topological quantum computing used \textquotedblleft knotted\textquotedblright{} quantum states of topological phases…
We describe rules for computing a homology theory of knots and links in $\mathbb{R}^3$. It is derived from the theory of framed BPS states bound to domain walls separating two-dimensional Landau-Ginzburg models with (2,2) supersymmetry. We…
Few physical systems with topologies more complicated than simple gaussian linking have been explored in detail. Here we focus on examples with higher topologies in non-relativistic quantum mechanics and in QCD.
Data science offers a powerful tool to understand objects in multiple sciences. In this paper we utilize concept of data science, most notably topological data analysis, to extend our understanding of knot theory. This approach provides a…
We give a new, elementary proof that Khovanov homology with $\mathbb{Z}/2\mathbb{Z}$--coefficients is invariant under Conway mutation. This proof also gives a strategy to prove Baldwin and Levine's conjecture that $\delta$--graded knot…
Knots and links represent a fundamental motif of non-local connectivity that permeates the physical sciences from string theory to protein folds. While spectral braiding has been explored in two-band non-Hermitian models across various…
In this paper, we introduce the annular instanton Floer homology which is defined for links in a thickened annulus. It is an analogue of the annular Khovanov homology. A spectral sequence whose second page is the annular Khovanov homology…
We introduce and explore the relation between knot invariants and quiver representation theory, which follows from the identification of quiver quantum mechanics in D-brane systems representing knots. We identify various structural…