arXiv subjects

Benjamin Burton

Publications and source records attributed to Benjamin Burton.

6 recordsLinked to original sources

arXiv2026

Triangulating Spun 2-Knot Complements

A $2$-knot is an embedding of a $2$-sphere into the $4$-sphere. Similar to the case of embedding circles into the $3$-sphere, this allows the $2$-sphere to be knotted. In this short paper, we present an algorithm to generate triangulations of the exteriors of $2$-knots obtained by spinning $1$-knots. We give an implementation of the algorithm in \emph{Regina} and present triangulations of exteriors obtained from all $1$-knots with up to eight crossings.

math.GT↗

arXiv2019

Knot Diagrams of Treewidth Two

In this paper, we study knot diagrams for which the underlying graph has treewidth two. We give a linear time algorithm for the following problem: given a knot diagram of treewidth two, does it represent the unknot? We also show that for a link diagram of treewidth two we can test in linear time if it represents the unlink. From the algorithm, it follows that a diagram of the unknot of treewidth 2 can always be reduced to the trivial diagram with at most $n$ (un)twist and (un)poke Reidemeister moves.

cs.DS↗

arXiv2018

The parameterized complexity of finding a 2-sphere in a simplicial complex

We consider the problem of finding a subcomplex K' of a simplicial complex K such that K' is homeomorphic to the 2-dimensional sphere, S^2. We study two variants of this problem. The first asks if there exists such a K' with at most k triangles, and we show that this variant is W[1]-hard and, assuming ETH, admits no O(n^{o(sqrt(k))}) time algorithm. We also give an algorithm that is tight with regards to this lower bound. The second problem is the dual of the first, and asks if K' can be found by removing at most k triangles from K. This variant has an immediate O(3^k poly(|K|)) time algorithm, and we show that it admits a polynomial kernelization to O(k^2) triangles, as well as a polynomial compression to a weighted version with bit-size O(k log k).

cs.DS↗

arXiv2015

Combinatorial Seifert fibred spaces with transitive cyclic automorphism group

In combinatorial topology we aim to triangulate manifolds such that their topological properties are reflected in the combinatorial structure of their description. Here, we give a combinatorial criterion on when exactly triangulations of 3-manifolds with transitive cyclic symmetry can be generalised to an infinite family of such triangulations with similarly strong combinatorial properties. In particular, we construct triangulations of Seifert fibred spaces with transitive cyclic symmetry where the symmetry preserves the fibres and acts non-trivially on the homology of the spaces. The triangulations include the Brieskorn homology spheres $Σ(p,q,r)$, the lens spaces $\operatorname{L} (q,1)$ and, as a limit case, $(\mathbf{S}^2 \times \mathbf{S}^1)^{\# (p-1)(q-1)}$.

math.GT↗

arXiv2006

Ferroics: magnetic-compass lattice and optical phonon dispersions of dipolar crystals

We report a simple safe and attractive pedagogic demonstration with magnetic compasses that facilitates an intuitive understanding of the concept that ferromagnetism and ferroelectricity do not result from dipole-dipole interactions alone. Phonon dispersion relations were calculated for a 3-dimensional simple-cubic dipole crystal and a 2-dimensional square-lattice dipole crystal. The latter calculation corresponds to the compass demonstration that confirm the antiferro ground state structure. A sum rule for the three optical phonon frequencies is discussed. A mathematical non-analyticity in the longitudinal optical phonon are also illustrated.

cond-mat.mtrl-sci↗

arXiv2002

First Principles Studies of KNbO$_3$, KTaO$_3$ and LiTaO$_3$ Solid Solutions

KTaO_3-based solid solutions exhibit a variety of interesting physical phenomena. To better understand these phenomena, we performed first-principles calculations on [K_{1-x}, Li_x]TaO_3 (KLT) and K[Ta_{1-x}, Nb_x]O_3 (KTN) supercells. Our results show Li displacements and potential barrier heights in KLT that are in excellent agreement with values obtained from experimental fits. Dramatic changes in B-site dynamical charges occur in KTN in response to changes in near neighbor (nn) coordination. These effects can be explained by heterogeneity in the local electronic dielectric permittivity.

cond-mat.mtrl-sci↗