Related papers: Tangle Machines II: Invariants
We develop the properties of the $n$-th sequential topological complexity $TC_n$, a homotopy invariant introduced by the third author as an extension of Farber's topological model for studying the complexity of motion planning algorithms in…
Transformers play a central role in the inner workings of large language models. We develop a mathematical framework for analyzing Transformers based on their interpretation as interacting particle systems, which reveals that clusters…
In this series of papers, we propose a theory of enumerative invariants counting self-dual objects in self-dual categories. Ordinary enumerative invariants in abelian categories can be seen as invariants for the structure group $\mathrm{GL}…
Two novel computing models based on an infinite tessellation of space-time are introduced. They consist of recursively coupled primitive building blocks. The first model is a scale-invariant generalization of cellular automata, whereas the…
The notion of a pseudoknot is defined as an equivalence class of knot diagrams that may be missing some crossing information. We provide here a topological invariant schema for pseudoknots and their relatives, 4-valent rigid vertex spatial…
Time-series classification is an important domain of machine learning and a plethora of methods have been developed for the task. In comparison to existing approaches, this study presents a novel method which decomposes a time-series…
Particle systems admit a variety of tensor product structures (TPSs) depending on the complete system of commuting observables chosen for the analysis. Different notions of entanglement are associated with these different TPSs. Global…
Recent formal classifications of crystalline topological insulators predict that the combination of time-reversal and rotational symmetry gives rise to topological invariants beyond the ones known for other lattice symmetries. Although the…
Given a countable set of sites and a collection of flip rates at each site, we give a sufficient condition on the long-range dependancies of the flip rates ensuring the well-definedness of the corresponding spin system. This hypothesis has…
In Part I of this series we described three algorithms that construct canonical tree-decompositions of graphs which distinguish all their k-blocks and tangles of order k. We now establish bounds on the number of parts in these…
Topological phases of matter are often understood and predicted with the help of crystal symmetries, although they don't rely on them to exist. In this chapter we review how topological phases have been recently shown to emerge in amorphous…
Consider a finite sequence of permutations of the elements 1,...,n, with the property that each element changes its position by at most 1 from any permutation to the next. We call such a sequence a tangle, and we define a move of element i…
Tensors play a pivotal role in the realms of science and engineering, particularly in the realms of data analysis, machine learning, and computational mathematics. The process of unfolding a tensor into matrices, commonly known as tensor…
Topological transforms are parametrized families of topological invariants, which, by analogy with transforms in signal processing, are much more discriminative than single measurements. The first two topological transforms to be defined…
Thurston defined invariant laminations, i.e. collections of chords of the unit circle $S^1$ (called \emph{leaves}) that are pairwise disjoint inside the open unit disk and satisfy a few dynamical properties. To be directly associated to a…
We define invariants of words in arbitrary groups, measuring how letters in a word are interleaving, perfectly detecting the dimension series of a group. These are the letter-braiding invariants. On free groups, braiding invariants coincide…
We present a unified framework to describe lattice gauge theories by means of tensor networks: this framework is efficient as it exploits the high amount of local symmetry content native of these systems describing only the gauge invariant…
A planar Tangle is a smooth simple closed curve piecewise defined by quadrants of circles with constant curvature. We can enumerate Tangles by counting their dual graphs, which consist of a certain family of polysticks. The number of…
The complexity of condensed matter arises from emergent behaviors that cannot be understood by analyzing individual constituents in isolation. While traditional condensed-matter approaches-developed primarily for ideal crystalline…
By a tensor we mean an element of a tensor product of vector spaces over a field. Up to a choice of bases in factors of tensor products, every tensor may be coordinatized, that is, represented as an array consisting of numbers. This note is…