\documentclass{amsart}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amscd}
\usepackage{latexsym}
\usepackage{url}
\usepackage{graphicx}
\usepackage{pinlabel}
\usepackage{lineno}
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{cor}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem*{question}{Question}
\newtheorem*{definition}{Definition}
\newtheorem{notitle}{ }
\theoremstyle{definition}
%\newtheorem{definition}{Definition}
\newtheorem*{conj}{Conjecture}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newcommand{\T}{\mathcal{T}_K}
\newcommand{\TT}{\mathcal{T}_{K_i}}
\newcommand{\spinc}{$spin^{\text{c}}$ }
\parskip.06in
\begin{document}
\baselineskip.5cm
\title {Building an atlas of tangles}
\author{Kenneth L Baker}
\address{Kenneth L Baker, Assistant Professor, Department of Mathematics, Univeristy of Miami, Miami, FL}
\email{\rm{kenken@math.miami.edu}}
\author{Dror Bar-Natan}
\address{Dror Bar-Natan, Professor, Department of Mathematics, University of Toronto, Toronto, Ontario, Canada}
\email{\rm{drorbn@math.toronto.edu}}
\author{Neil Hoffman}
\address{Neil Hoffman, Post doctoral researcher, Department of Mathematics and Statistics, University of Melbourne,
Melbourne, Australia}
\email{\rm{nhoffman@ms.unimelb.edu.au}}
\author{Anastasiia Tsvietkova}
\address{Anastasiia Tsvietkova, Post doctoral researcher, ICERM, Brown University, Providence, RI}
\email{\rm{n.tsvet@gmail.com}}
\maketitle
\linenumbers
%The study of low dimensional topology has been greatly advanced
%by a wealth of simple examples to analyze. The examples have been
%collated into tables (see \cite{knotatlas}), which
%have been particularly
%helpful to the study of knots and links. For example, when building a
%new invariant one can test if this invariant distinguishes pairs of knots
%that might be hard to prove are non-isotopic.
The
fields of topology and geometry have been greatly advanced by a wealth
of simple examples to analyze.
In the study of 3-manifolds from
topological, geometric, combinatorial and quantum points of view, the
first examples that allow to check intuitive ideas are often knots and
links.
The purpose of this
research project is to extend the classification of knots and links to a
classification of tangles, which one can consider as building blocks
for knots and links.
While \emph{tangle} can describe a variety of objects, we will consider tangles
comprised of $a$ arcs and $b$ circles properly embedded in the $3$--ball up to
isotopy fixing the boundary of $a$ in the boundary of the ball.
% We will use $(B^3,T)$ to denote such an object.
First, we seek to
enumerate all tangles such that $a=2$ of small complexity, namely that the
tangle admits a planar diagram having $m$ or fewer crossings. (The actual $m$ will be
determined later by the computation time of our algorithms.) Second, we will
classify these tangles up to isotopy type. In future work, we would seek to
develop and apply techniques to higher values for $a$.
The results of this project should be of interest to (at least) two audiences.
First, 3-manifold topologists, especially knot theorists, will benefit from having this
``infrastructure project'' addressed. By providing a concise and defined table
to test ideas against, topologists will be able to understand their relevance and to gauge their
computability of any new tangle invariant. There are numerous papers
where this analysis has
been applied to the tabulations of knots and links (see
papers citing \cite{hoste1998first} and Rolfsen's table \cite{rolfsen}).
Tangles themselves form the building blocks for knots and
links, and so a deeper understanding of tangles would lead to advances
in the field.
A second
audience for this project is applied mathematicians, such as, for
example, DNA topologists.
The double helix backbone of DNA may be modeled with tangles and its topology effects its various functions, replication among others (see e.g. \cite{bakerbuck}). The action of certain proteins upon DNA may be studied in this model through {\em subtangle replacements} \cite{ernst-sumners}, i.e. removing a rational tangle from a link or tangle and closing
that tangle with a second rational tangle.
%DNA is naturally
%tangled in the cell, however, the topology of the DNA as an embedded tangle is
%one factor that affects how a cell can replicate as well as a variety of other
%interesting behavior (see \cite{buckflapan} for example). Often, DNA topologists
%are interested in how a tangle behaves under \emph{tangle filling}, i.e. closing
%of a tangle with $a$ arcs embedded in a 3-ball with a second tangle having
%$a$ arcs in a 3-ball.
Our research program is straightforward, however we will now provide greater detail
to what is mentioned above. First, we seek to extend and implement the enumeration of knots
and links as performed in the tables of \cite{knotatlas} to our setting in order to enumerate all possible tangles
with small crossing number comprised of two arcs and $n$ circles.
Second, we hope to classify all such tangles up to isotopy.
Initially, there are two techniques we hope to use to distinguish these tangles.
In recent work (see \cite[$\S$4.2]{ATThesis}), the fourth investigator applies the
idea of \emph{encircled tangles}, namely adding an extra embedded
circle to a tangle diagram such this embedded circle alternates between over and under crossings
with the two arcs of the tangle that connect to the boundary. Excluding a few
pathological cases,
the tangle inside the circle has a canonical hyperbolic %used canonical instead of unique.
structure independent of the link containing it by \cite[Theorem 7.5]{ATThesis}. Therefore,
this structure can be used as an invariant to distinguish tangles.
%this encircled tangle
%has a hyperbolic structure by \cite[Theorem 4.2.3]{ATThesis}, and so this structure can be used as a
%tangle invariant.
Additionally, to any tangle in the ball we can associate a $3$--manifold that is a double cover of the ball branched over the arcs and circles of the tangle. In particular, when the tangle has only two arcs, this manifold will have torus boundary. A wealth of $3$--manifold invariants may then be deployed for distinguishing non-homeomorphic tangles.
%
%Additionally, to any tangle $(B^3,T)$, we can associate a double branched cover $M$ such that there
%is a continuous map $f: M \rightarrow (B^3,T)$ that is two-to-one between $M-f^{-1}(T)$ and $B^3-T$
%and one-to-one on $f^{-1}(T)$ and $T$. This double branched cover will be a 3-manifold, and so
%invariants of $M$ can be applied in distinguishing the homeomorphism type of $(B^3,T)$.
%Furthermore,
%in the case that $T$ has only two arcs $M$ will have a torus boundary (see \cite{walsh}
%for example) and
%so more refined geometric invariants such has hyperbolic volume associated to $M$
%can be employed as
%part of the identification.
While some ideas along these lines have been implemented as part of the
software ORB \cite{orb}, that piece of software is no longer being updated. In addition to the
updates for compatibility with current computer architectures, this software would also
need to be tweaked to include a scriptable interface.
The SQuaREs model is well suited for building such an atlas of tangles. This project will require
a considerable use of computational tools. Having a number of people working
in collaboration to first build and then later maintain and grow the atlas seems ideal for the
work that needs to be done. In addition, the final product will be freely available software that would include a sortable repository
similar to \cite{knotatlas}.
% Thus, drawing on the ideas and inputs from each of four authors will
%ultimately lead to a more flexible interface that will provide maximum utility to the user.
The third and fourth authors will take
responsibility for the large scale computational tasks and maintenance of the atlas, while the second author, based on his experience with \cite{knotatlas}, and the first author drawing on his experiences with both 3-manifold topology and DNA topology \cite{bakerbuck, baker}, will guide the project ensuring the resulting atlas and its interface are flexible and user-friendly.
In addition, our team meets the criterion of the SQuaREs model. This collaboration would bring
together not only four people at different stages of their careers, namely two post doctoral researchers and two professors,
but also four people that are
well suited to solve this specific problem because their variety of expertise.
%Since
%our audience is both 3-manifold topologists and biologist involved understanding applied
%problems relating to DNA, this team seems ideally suited for these tasks.
\newpage
\begin{thebibliography}{1}
\bibitem{knotatlas} Dror Bar-Natan et. al, The knot atlas, \url{http://katlas.org}.
\bibitem{bakerbuck} Kenneth L Baker and Dorothy Buck
\emph{The classification of rational subtangle replacements between rational tangles}, Alg. Geom. Top. 13 (2013) 1413--1463.
\bibitem{baker} Kenneth L Baker {\it Site specific recombination on unknot and unlink substrates producing two-bridge links. (Survey)},
in N. Jonoska, M. Saito (Eds.), "Discrete and Topological Models in Molecular Biology", Springer, Natural Computing Series, 2013, ISBN 978-3-642-40192-3.
\bibitem{buckflapan} Dorothy Buck and Eric Flapan {\em Applications of Knot Theory }, AMS, 2009.
\bibitem{ernst-sumners} Claus Ernst, De Witt Sumners, {\em A calculus for rational tangles: applications to DNA recombination}, Math. Proc. Camb. Phil.
Soc., 108, (1990) 409--515.
\bibitem{orb} Damian Heard, Orb, based on Jeff Weeks' Snappea kernel, \url{http://www.ms.unimelb.edu.au/~snap/orb.html}.
\bibitem{hoste1998first} Jim Hoste and Morwen Thistlethwaite and Jeff Weeks {\em The first 1,701,936 knots} The Mathematical Intelligencer, 20(4) (1998) 33-48.
\bibitem{rolfsen} Dale Rolfsen {\em Knots and lins}, AMS Chelsea Pub. 2003.
\bibitem{ATThesis} Morwen Thistlethwaite and Anastasiia Tsvietkova \emph{Hyperbolic Structures from Link Diagrams}, ProQuest LLC 2012.
\emph{An alternative approach to
hyperbolic structures on link complements}, to appear in Alg. Geom. Top., 1--32.
\bibitem{walsh} Genevieve Walsh {\em Orbifolds and commensurabilty}, ``Interactions Between Hyperbolic Geometry, Quantum Topology and Number Theory" Contemporary Mathematics Vol 541, (2011) 221--231.
\end{thebibliography}
\end{document}