Homological Error Correction: Classical and Quantum Codes

We prove several theorems characterizing the existence of homological error correction codes both classically and quantumly. Not every classical code is homological, but we find a family of classical homological codes saturating the Hamming bound. In the quantum case, we show that for non-orientable surfaces it is impossible to construct homological codes based on qudits of dimension $D>2$, while for orientable surfaces with boundaries it is possible to construct them for arbitrary dimension $D$. We give a method to obtain planar homological codes based on the construction of quantum codes on compact surfaces without boundaries. We show how the original Shor's 9-qubit code can be visualized as a homological quantum code. We study the problem of constructing quantum codes with optimal encoding rate. In the particular case of toric codes we construct an optimal family and give an explicit proof of its optimality. For homological quantum codes on surfaces of arbitrary genus we also construct a family of codes asymptotically attaining the maximum possible encoding rate. We provide the tools of homology group theory for graphs embedded on surfaces in a self-contained manner.Comment: Revtex4 fil

Chord Diagrams and Coxeter Links

This paper presents a construction of fibered links $(K,\Sigma)$ out of chord diagrams \sL. Let $\Gamma$ be the incidence graph of \sL. Under certain conditions on \sL the symmetrized Seifert matrix of $(K,\Sigma)$ equals the bilinear form of the simply-laced Coxeter system $(W,S)$ associated to $\Gamma$; and the monodromy of $(K,\Sigma)$ equals minus the Coxeter element of $(W,S)$. Lehmer's problem is solved for the monodromy of these Coxeter links.Comment: 18 figure

The complement of proper power graphs of finite groups

For a finite group $G$, the proper power graph $\mathscr{P}^*(G)$ of $G$ is the graph whose vertices are non-trivial elements of $G$ and two vertices $u$ and $v$ are adjacent if and only if $u \neq v$ and $u^m=v$ or $v^m=u$ for some positive integer $m$. In this paper, we consider the complement of $\mathscr{P}^*(G)$, denoted by ${\overline{\mathscr{P}^*(G)}}$. We classify all finite groups whose complement of proper power graphs is complete, bipartite, a path, a cycle, a star, claw-free, triangle-free, disconnected, planar, outer-planar, toroidal, or projective. Among the other results, we also determine the diameter and girth of the complement of proper power graphs of finite groups.Comment: 29 pages, 14 figures, Lemma 4.1 has been added and consequent changes have been mad

Book embeddings of Reeb graphs

Let $X$ be a simplicial complex with a piecewise linear function $f:X\to\mathbb{R}$. The Reeb graph $Reeb(f,X)$ is the quotient of $X$, where we collapse each connected component of $f^{-1}(t)$ to a single point. Let the nodes of $Reeb(f,X)$ be all homologically critical points where any homology of the corresponding component of the level set $f^{-1}(t)$ changes. Then we can label every arc of $Reeb(f,X)$ with the Betti numbers $(\beta_1,\beta_2,\dots,\beta_d)$ of the corresponding $d$-dimensional component of a level set. The homology labels give more information about the original complex $X$ than the classical Reeb graph. We describe a canonical embedding of a Reeb graph into a multi-page book (a star cross a line) and give a unique linear code of this book embedding.Comment: 12 pages, 5 figures, more examples will be at http://kurlin.or

Developing a Mathematical Model for Bobbin Lace

Bobbin lace is a fibre art form in which intricate and delicate patterns are created by braiding together many threads. An overview of how bobbin lace is made is presented and illustrated with a simple, traditional bookmark design. Research on the topology of textiles and braid theory form a base for the current work and is briefly summarized. We define a new mathematical model that supports the enumeration and generation of bobbin lace patterns using an intelligent combinatorial search. Results of this new approach are presented and, by comparison to existing bobbin lace patterns, it is demonstrated that this model reveals new patterns that have never been seen before. Finally, we apply our new patterns to an original bookmark design and propose future areas for exploration.Comment: 20 pages, 18 figures, intended audience includes Artists as well as Computer Scientists and Mathematician