191 research outputs found
Hamming distance kernelisation via topological quantum computation
We present a novel approach to computing Hamming distance and its kernelisation within Topological Quantum Computation. This approach is based on an encoding of two binary strings into a topological Hilbert space, whose inner product yields a natural Hamming distance kernel on the two strings. Kernelisation forges a link with the field of Machine Learning, particularly in relation to binary classifiers such as the Support Vector Machine (SVM). This makes our approach of potential interest to the quantum machine learning community
On Upper Bounds for Toroidal Mosaic Numbers
In this paper, we work to construct mosaic representations of knots on the
torus, rather than in the plane. This consists of a particular choice of the
ambient group, as well as different definitions of contiguous and suitably
connected. We present conditions under which mosaic numbers might decrease by
this projection, and present a tool to measure this reduction. We show that the
order of edge identification in construction of the torus sometimes yields
different resultant knots from a given mosaic when reversed. Additionally, in
the Appendix we give the catalog of all 2 by 2 torus mosaics.Comment: 10 pages, 111 figure
Knot Theory: from Fox 3-colorings of links to Yang-Baxter homology and Khovanov homology
This paper is an extended account of my "Introductory Plenary talk at Knots
in Hellas 2016" conference We start from the short introduction to Knot Theory
from the historical perspective, starting from Heraclas text (the first century
AD), mentioning R.Llull (1232-1315), A.Kircher (1602-1680), Leibniz idea of
Geometria Situs (1679), and J.B.Listing (student of Gauss) work of 1847. We
spend some space on Ralph H. Fox (1913-1973) elementary introduction to diagram
colorings (1956). In the second section we describe how Fox work was
generalized to distributive colorings (racks and quandles) and eventually in
the work of Jones and Turaev to link invariants via Yang-Baxter operators, here
the importance of statistical mechanics to topology will be mentioned. Finally
we describe recent developments which started with Mikhail Khovanov work on
categorification of the Jones polynomial. By analogy to Khovanov homology we
build homology of distributive structures (including homology of Fox colorings)
and generalize it to homology of Yang-Baxter operators. We speculate, with
supporting evidence, on co-cycle invariants of knots coming from Yang-Baxter
homology. Here the work of Fenn-Rourke-Sanderson (geometric realization of
pre-cubic sets of link diagrams) and Carter-Kamada-Saito (co-cycle invariants
of links) will be discussed and expanded.
Dedicated to Lou Kauffman for his 70th birthday.Comment: 35 pages, 31 figures, for Knots in Hellas II Proceedings, Springer,
part of the series Proceedings in Mathematics & Statistics (PROMS
On a Subposet of the Tamari Lattice
We explore some of the properties of a subposet of the Tamari lattice
introduced by Pallo, which we call the comb poset. We show that three binary
functions that are not well-behaved in the Tamari lattice are remarkably
well-behaved within an interval of the comb poset: rotation distance, meets and
joins, and the common parse words function for a pair of trees. We relate this
poset to a partial order on the symmetric group studied by Edelman.Comment: 21 page
The Combinatorics of Alternating Tangles: from theory to computerized enumeration
We study the enumeration of alternating links and tangles, considered up to
topological (flype) equivalences. A weight is given to each connected
component, and in particular the limit yields information about
(alternating) knots. Using a finite renormalization scheme for an associated
matrix model, we first reduce the task to that of enumerating planar
tetravalent diagrams with two types of vertices (self-intersections and
tangencies), where now the subtle issue of topological equivalences has been
eliminated. The number of such diagrams with vertices scales as for
. We next show how to efficiently enumerate these diagrams (in time
) by using a transfer matrix method. We give results for various
generating functions up to 22 crossings. We then comment on their large-order
asymptotic behavior.Comment: proceedings European Summer School St-Petersburg 200
Knots and Particles
Using methods of high performance computing, we have found indications that
knotlike structures appear as stable finite energy solitons in a realistic 3+1
dimensional model. We have explicitly simulated the unknot and trefoil
configurations, and our results suggest that all torus knots appear as
solitons. Our observations open new theoretical possibilities in scenarios
where stringlike structures appear, including physics of fundamental
interactions and early universe cosmology. In nematic liquid crystals and 3He
superfluids such knotted solitons might actually be observed.Comment: 9 pages, 4 color eps figures and one b/w because of size limit (color
version available from authors
Response to arXiv:0811.3876 "Comment on a recent conjectured solution of the three dimensional Ising model" by Wu et al
This is a Response to a recent Comment [F.Y. Wu et al., Phil. Mag. 88, 3093
(2008), arXiv:0811.3876] on the conjectured solution of the three-dimensional
(3D) Ising model [Z.D. Zhang, Phil. Mag. 87, 5309 (2007), arXiv:0705.1045].
Several points are made: 1) Conjecture 1, regarding the additional rotation, is
understood as performing a transformation for smoothing all the crossings of
the knots; 2) The weight factors in Conjecture 2 are interpreted as a novel
topologic phase; 3) The conjectured solution and its low- and high-temperature
expansions are supported by the mathematical theorems for the analytical
behavior of the Ising model. The physics behind the extra dimension is also
discussed briefly.Comment: 11 pages, 0 figure
Knot invariants in lens spaces
In this survey we summarize results regarding the Kauffman bracket, HOMFLYPT,
Kauffman 2-variable and Dubrovnik skein modules, and the Alexander polynomial
of links in lens spaces, which we represent as mixed link diagrams. These
invariants generalize the corresponding knot polynomials in the classical case.
We compare the invariants by means of the ability to distinguish between some
difficult cases of knots with certain symmetries
Canonical quantization of non-commutative holonomies in 2+1 loop quantum gravity
In this work we investigate the canonical quantization of 2+1 gravity with
cosmological constant in the canonical framework of loop quantum
gravity. The unconstrained phase space of gravity in 2+1 dimensions is
coordinatized by an SU(2) connection and the canonically conjugate triad
field . A natural regularization of the constraints of 2+1 gravity can be
defined in terms of the holonomies of . As a first step
towards the quantization of these constraints we study the canonical
quantization of the holonomy of the connection on the
kinematical Hilbert space of loop quantum gravity. The holonomy operator
associated to a given path acts non trivially on spin network links that are
transversal to the path (a crossing). We provide an explicit construction of
the quantum holonomy operator. In particular, we exhibit a close relationship
between the action of the quantum holonomy at a crossing and Kauffman's
q-deformed crossing identity. The crucial difference is that (being an operator
acting on the kinematical Hilbert space of LQG) the result is completely
described in terms of standard SU(2) spin network states (in contrast to
q-deformed spin networks in Kauffman's identity). We discuss the possible
implications of our result.Comment: 19 pages, references added. Published versio
Perfect k-Colored Matchings and (k+2)-Gonal Tilings
We derive a simple bijection between geometric plane perfect matchings on 2n points in convex position and triangulations on n+2 points in convex position. We then extend this bijection to monochromatic plane perfect matchings on periodically k-colored vertices and (k+2)-gonal tilings of convex point sets. These structures are related to a generalization of Temperley–Lieb algebras and our bijections provide explicit one-to-one relations between matchings and tilings. Moreover, for a given element of one class, the corresponding element of the other class can be computed in linear time
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