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\begin{document}

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\section*{knot invariant}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{knot_theory}{}\paragraph*{{Knot theory}}\label{knot_theory}

[[!include knot theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{in_string_theory}{In string theory}\dotfill \pageref*{in_string_theory} \linebreak
\noindent\hyperlink{via_5brane_bps_states}{Via 5-brane BPS states}\dotfill \pageref*{via_5brane_bps_states} \linebreak
\noindent\hyperlink{via_dd_bound_states__monopoles}{Via D$p$/D$(p+2)$ bound states / monopoles}\dotfill \pageref*{via_dd_bound_states__monopoles} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \textbf{knot invariant} is map from [[isotopy]] [[equivalence class]]es of [[knots]] to any kind of structure you could imagine. These are helpful because it is often much easier to check that the structures one maps to (numbers, groups, etc.) are different than it is to check that knots are different. To define a knot invariant, it suffices to define its value on [[knot diagrams]] and check that this value is preserved under the [[Reidemeister moves]] (possibly with the exception of the first Reidemeister move, in the case of an invariant of [[framed knots]]).

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item [[crossing number]]
\item [[bridge number]]
\item [[unknotting number]]
\item [[colorability]]
\item [[knot group]]= the [[fundamental group]] of the complement of the knot.
\item [[quandles]] and [[racks]]
\item [[knot genus]]
\item [[Jones polynomial]]
\item [[HOMFLY polynomial]]
\item [[Alexander polynomial]]
\item [[Reshetikhin-Turaev invariant]]s for any object in a [[ribbon category]]
\item [[Khovanov homology]]
\item [[Kauffman bracket]] (an invariant of [[framed links]] closely related to the [[Jones polynomial]])

\end{itemize}
Many of these extend to [[link invariant|link invariants]] or have variants that depend on the knot being oriented.

\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

(\ldots{})

\hypertarget{in_string_theory}{}\subsubsection*{{In string theory}}\label{in_string_theory}

Knot invariants arising in [[string theory]]/[[M-theory]]:

\hypertarget{via_5brane_bps_states}{}\paragraph*{{Via 5-brane BPS states}}\label{via_5brane_bps_states}

Discussion of [[knot invariants]] in terms of [[BPS states]] of [[M5-branes]]:

\begin{itemize}%
\item [[Edward Witten]], \emph{Fivebranes and Knots} (\href{http://arxiv.org/abs/1101.3216}{arXiv:1101.3216})


\item [[Davide Gaiotto]], [[Edward Witten]], \emph{Knot Invariants from Four-Dimensional Gauge Theory}, Advances in Theoretical and Mathematical Physics, Volume 16 (2012) Number 3 (\href{https://arxiv.org/abs/1106.4789}{arxiv:1106.4789})


\item [[Sergei Gukov]], [[Marko Stošić]], \emph{Homological algebra of knots and BPS states} (\href{http://arxiv.org/abs/1112.0030}{arXiv:1112.0030})


\item Ross Elliot, [[Sergei Gukov]], \emph{Exceptional knot homology} (\href{http://arxiv.org/abs/1505.01635}{arXiv:1505.01635})


\item Satoshi Nawata, Alexei Oblomkov, \emph{Lectures on knot homology} (\href{http://arxiv.org/abs/1510.01795}{arXiv:1510.01795})



\end{itemize}
\hypertarget{via_dd_bound_states__monopoles}{}\paragraph*{{Via D$p$/D$(p+2)$ bound states / monopoles}}\label{via_dd_bound_states__monopoles}

Relation of [[Dp-D(p+2)-brane bound states]]/[[Yang-Mills monopoles]] to [[knot invariants]] via [[chord diagrams]]:

\begin{itemize}%
\item S. Ramgoolam, B. Spence, S. Thomas, Section 3.2 of: \emph{Resolving brane collapse with $1/N$ corrections in non-Abelian DBI}, Nucl. Phys. B703 (2004) 236-276 (\href{https://arxiv.org/abs/hep-th/0405256}{arxiv:hep-th/0405256})


\item S. McNamara, Constantinos Papageorgakis, S. Ramgoolam, B. Spence, Appendix A of: \emph{Finite $N$ effects on the collapse of fuzzy spheres}, JHEP 0605:060, 2006 (\href{https://arxiv.org/abs/hep-th/0512145}{arxiv:hep-th/0512145})


\item Constantinos Papageorgakis, p. 161-162 of: \emph{On matrix D-brane dynamics and fuzzy spheres}, 2006 ([[Papageorgakis06.pdf:file]])



\end{itemize}
category: knot theory

[[!redirects knot invariants]] [[!redirects Knot Invariant]] [[!redirects Knot Invariants]] [[!redirects Knot invariant]] [[!redirects Knot invariants]]



\end{document}