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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{Vassiliev invariant} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{knot_theory}{}\paragraph*{{Knot theory}}\label{knot_theory} [[!include knot theory - contents]] \hypertarget{vassiliev_knot_invariants}{}\section*{{Vassiliev knot invariants}}\label{vassiliev_knot_invariants} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{relation_to_chord_diagrams_and_weight_systems}{Relation to Chord diagrams and weight systems}\dotfill \pageref*{relation_to_chord_diagrams_and_weight_systems} \linebreak \noindent\hyperlink{RelationToHomologOfLoopSpacesOfConfigurationSpaces}{Relation to homology of loop spaces of configuration spaces of points}\dotfill \pageref*{RelationToHomologOfLoopSpacesOfConfigurationSpaces} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak \noindent\hyperlink{relation_to_jones_polynomial}{Relation to Jones polynomial:}\dotfill \pageref*{relation_to_jones_polynomial} \linebreak \noindent\hyperlink{relation_to_homology_of_loop_spaces_of_configuration_spaces}{Relation to homology of loop spaces of configuration spaces}\dotfill \pageref*{relation_to_homology_of_loop_spaces_of_configuration_spaces} \linebreak \noindent\hyperlink{as_chernsimons_amplitudes}{As Chern-Simons amplitudes}\dotfill \pageref*{as_chernsimons_amplitudes} \linebreak \noindent\hyperlink{vassiliev_invariants_of_braids}{Vassiliev invariants of braids}\dotfill \pageref*{vassiliev_invariants_of_braids} \linebreak \noindent\hyperlink{general_2}{General}\dotfill \pageref*{general_2} \linebreak \noindent\hyperlink{as_observables_on_fuzzy_spheres}{As observables on fuzzy spheres}\dotfill \pageref*{as_observables_on_fuzzy_spheres} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The space of [[knots]] in the [[Euclidean space]] $\mathbb{R}^3$ (or in the [[3-sphere]] $S^3$) is an [[open subset|open]] [[submanifold]] of the [[smooth loop space]]. [[knot invariant|Knot invariants]] are [[locally constant functions]] on this manifold. The [[complement]] of the space of knots is called the [[discriminant]] and consists of all [[singular knots]]. If we consider those singular knots with only a finite number of double points, we can build a [[cubical complex]] from this data. The vertices in the complex are labelled by the isotopy classes of knots, and more generally the $n$-cubes by the isotopy classes of singular knots with $n$ double points (and a few other technical pieces of information). The boundary operator resolves a double crossing either upwards or downwards according to the orientation at the crossing. A \textbf{Vassiliev invariant} is simply a cubical morphism from this complex to an abelian group that vanishes above a certain degree. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} One does not need the language of cubical complexes to \emph{define} Vassiliev invariants. Rather, there is a general method whereby a [[knot invariant]] can be extended to all [[singular knots]] with only finitely many double points (and no other singularities) using the [[Vassiliev skein relations]]. \begin{defn} \label{vinv}\hypertarget{vinv}{} A \textbf{Vassiliev invariant} of degree (or order) $\le n$ is a knot invariant whose extension to singular knots (with double points) vanishes on all singular knots with more than $n$ double points. \end{defn} As is standard, it is of degree $n$ if it is of degree $\le n$ but not $\le n - 1$. Vassiliev invariants are also called \textbf{finite type invariants}. \begin{remark} \label{}\hypertarget{}{} The degree of Vassiliev invariants defines a filtration on the space of knots (and more particularly, on the [[algebra of knots]]). Two knots are $n$-equivalent if all the Vassiliev invariants of degree $\le n$ agree on them. In particular, a knot that is $n$-equivalent to the unknot is said to be $n$-trivial. \end{remark} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{relation_to_chord_diagrams_and_weight_systems}{}\subsubsection*{{Relation to Chord diagrams and weight systems}}\label{relation_to_chord_diagrams_and_weight_systems} A function which is constant on nonsingular knots may be extended to a Vassiliev invariant of degree 0 by applying the [[Vassiliev skein relations]], and conversely, any Vassiliev invariant of degree 0 must be constant on nonsingular knots. Likewise, any Vassiliev invariant of degree 1 must be constant on nonsingular knots. Any [[singular knot]] $f : S^1 \to \mathbb{R}^3$ with $n$ distinct double points $x_1,\dots,x_n \in \mathbb{R}^3$ gives rise to a [[chord diagram]] of order $n$, consisting of the circle $S^1$ with a chord connecting each pair of points $f^{-1}(x_1), \dots, f^{-1}(x_n)$. The importance of this construction for singular knots comes from the fact that any finite type invariant determines a function on chord diagrams: \begin{utheorem} Let $v$ be a Vassiliev invariant of degree $\le n$. Then the value of $v$ on a singular knot with $n$ distinct double points depends only on the chord diagram of the knot, and not on the knot itself. \end{utheorem} Conversely, one can ask which functions on chord diagrams come from finite type invariants. The answer is that Vassiliev invariants (of degree $\le n$) can essentially be identified with \emph{weight systems} (of order $n$), which are functions on chord diagrams (of order $n$) satisfying two properties called the ``1-term relation'' (or ``framing independence'') and the ``4-term relation'': see Theorem 1 of \hyperlink{BarNatan95}{Bar-Natan 95} (or Theorem 6.2.13 of \hyperlink{LandoZvonkin}{Lando \& Zvonkin}). \hypertarget{RelationToHomologOfLoopSpacesOfConfigurationSpaces}{}\subsubsection*{{Relation to homology of loop spaces of configuration spaces of points}}\label{RelationToHomologOfLoopSpacesOfConfigurationSpaces} We discuss the relation between Vassiliev invariants and the [[Euler characteristic]] of the [[ordinary homology]] of [[loop spaces]] of [[configuration spaces of points]]: For $n, q \in \mathbb{N}$ and $q \geq 1$, write \begin{enumerate}% \item $Conf_n\big( \mathbb{R}^{q+2} \big)$ for the [[configuration space of points|configuration space of n ordered points]] in [[Euclidean space]] $mathbb{R}^{q+2}$; \item $\Omega Conf_n\big( \mathbb{R}^{q+2} \big)$ for the corresponding [[based loop space]] (for any choice of base point); \item $H_\bullet\Big(\Omega Conf_n\big( \mathbb{R}^{q+2} \big), \mathbb{C} \Big)$ for the [[ordinary homology]] of this loop space, with [[coefficients]] in the [[complex numbers]]; \item $\chi H_\bullet\Big(\Omega Conf_n\big( \mathbb{R}^{q+2} \big), \mathbb{C} \Big)$ for the [[Euler characteristic]]-series of the homology \end{enumerate} Write also \begin{enumerate}% \item $V^n_k$ for the [[complex vector space]] of [[Vassiliev invariants]] of order $k$ for [[pure braids]] with $n$ strands;; \item $A^n_k$ for the [[complex vector space]] spanned by the horizontal [[chord diagrams]] with $n$ vertical strands modulo the ``horizontal 4T relation'' \end{enumerate} such that there is an [[linear isomorphism]] \begin{displaymath} V^n_k/V^n_{k-1} \simeq (A^n_k)^\ast \end{displaymath} between the [[quotient vector space]] of [[Vassiliev invariants]] and the [[dual vector space]] of [[chord diagrams]]. Then: The [[Euler characteristic]]-series (\ldots{}) of the homology of the loop spaces of configuration spaces \begin{displaymath} \chi H_\bullet\Big(\Omega Conf_n\big( \mathbb{R}^{q+2} \big), \mathbb{C} \Big) \;=\; \Big[ \big( 1 - t^q \big) \cdot \big( 1 - 2 t^q \big) \cdots \big( 1 - (n-1) t^q \big) \Big]^{-1} \end{displaymath} and is related to the complex [[dimensions]] of spaces of Vassiliev invariants according to \begin{displaymath} \chi H_\bullet \Big( \Omega Conf_n\big( \mathbb{R}^{3} \big), \mathbb{C} \Big) \;=\; \underset{k \in \mathbb{N}}{\sum} dim_{\mathbb{C}}\big( A^n_k \big) t^k \end{displaymath} (\hyperlink{CohenGitler01}{Cohen-Gitler 01, Prop. 9.1}, based on \hyperlink{Cohen76}{Cohen 76} and \hyperlink{Kohno94}{Kohno 94}) \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \begin{enumerate}% \item The $n$th coefficient of the [[Conway polynomial]] is a Vassiliev invariant of order $\le n$. \end{enumerate} (\ldots{}) \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[knot theory]] \item [[Jones polynomial]] \item [[Kontsevich integral]] \item [[singularity]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{general}{}\subsubsection*{{General}}\label{general} The original articles are \begin{itemize}% \item [[Viktor Vassiliev]], \emph{Complements of discriminants of smooth maps: topology and applications}, Amer. Math. Soc. 1992. \item [[Maxim Kontsevich]], \emph{Vassiliev's knot invariants}, Advances in Soviet Mathematics, Volume 16, Part 2, 1993 (\href{http://pagesperso.ihes.fr/~maxim/TEXTS/VassilievKnot.pdf}{pdf}) \end{itemize} Review: \begin{itemize}% \item [[Dror Bar-Natan]], \emph{On the Vassiliev knot invariants}, Topology Volume 34, Issue 2, April 1995, Pages 423-472 (\href{http://www.math.toronto.edu/~drorbn/papers/OnVassiliev/}{web}, ) \item S. Chmutov, S. Duzhin, J. Mostovoy, \emph{Introduction to Vassiliev knot invariants} (\href{http://arxiv.org/abs/1103.5628}{arxiv/1103.5628}) \item Sergei K. Lando and Alexander K. Zvonkin, Chapter 6 of: \emph{Graphs on Surfaces and Their Applications}, Springer, 2004. \end{itemize} More literature is listed at \begin{itemize}% \item [[Dror Bar-Natan]], Sergei Duzhin, \emph{\href{http://www.pdmi.ras.ru/~duzhin/VasBib/Long}{Bibliography of Vassiliev Invariants}} \end{itemize} See also \begin{itemize}% \item Mathworld, \emph{\href{http://mathworld.wolfram.com/VassilievInvariant.html}{Vassiliev invariant}} \end{itemize} Concrete computations: \begin{itemize}% \item Jan Kneissler, \emph{On spaces of connected graphs I: Properties of Ladders}, Proc. Internat. Conf. ``Knots in Hellas `98'', Series on Knots and Everything, vol. 24 (2000), 252-273 (\href{https://arxiv.org/abs/math/0301018}{arXiv:math/0301018}) \item Jan Kneissler, \emph{On spaces of connected graphs II: Relations in the algebra Lambda}, Jour. of Knot Theory and its Ramif. vol. 10, no. 5 (2001), 667-674 (\href{https://arxiv.org/abs/math/0301019}{arXiv:math/0301019}) \item Jan Kneissler, \emph{On spaces of connected graphs III: The Ladder Filtration}, Jour. of Knot Theory and its Ramif. vol. 10, no. 5 (2001), 675-686 (\href{https://arxiv.org/abs/math/0301020}{arXiv:math/0301020}) \item [[Pierre Vogel]], \emph{Algebraic structures on modules of diagrams}, Journal of Pure and Applied Algebra Volume 215, Issue 6, June 2011, Pages 1292-1339 (\href{https://webusers.imj-prg.fr/~pierre.vogel/diagrams.pdf}{pdf}) \end{itemize} \hypertarget{relation_to_jones_polynomial}{}\subsubsection*{{Relation to Jones polynomial:}}\label{relation_to_jones_polynomial} Relation to the [[Jones polynomial]]: \begin{itemize}% \item Joan S. Birman; Xiao-Song Lin, \emph{Knot polynomials and Vassiliev's invariants}, Inventiones mathematicae (1993) Volume: 111, Issue: 2, page 225-270 (\href{https://eudml.org/doc/144077}{https://dml:144077}) \end{itemize} Relation to other polynomial [[knot invariants]]: \begin{itemize}% \item Myeong-Ju Jeong, Chan-Young Park, \emph{Polynomial invariants and Vassiliev invariants}, Geom. Topol. Monogr. 4 (2002) 89-101 (\href{https://arxiv.org/abs/math/0211045}{arxiv:math/0211045}) \end{itemize} \hypertarget{relation_to_homology_of_loop_spaces_of_configuration_spaces}{}\subsubsection*{{Relation to homology of loop spaces of configuration spaces}}\label{relation_to_homology_of_loop_spaces_of_configuration_spaces} Relation to the [[Euler characteristic]] of the [[ordinary homology]] of [[loop spaces]] of [[configuration spaces of points]] \begin{itemize}% \item [[Fred Cohen]], [[Samuel Gitler]], \emph{Loop spaces of configuration spaces, braid-like groups, and knots}, In: Aguadé J., Broto C., [[Carles Casacuberta]] (eds.) \emph{Cohomological Methods in Homotopy Theory}. Progress in Mathematics, vol 196. Birkhäuser, Basel 2001 (\href{https://doi.org/10.1007/978-3-0348-8312-2_7}{doi:10.1007/978-3-0348-8312-2\_7}) \end{itemize} based on \begin{itemize}% \item [[Toshitake Kohno]], \emph{Vassiliev invariants and de Rham complex on the space of knots}, Contemporary Mathematics 179 (1994): 123-123. \item [[Fred Cohen]], \emph{The homology of $\mathcal{C}_{n+1}$-Spaces, $n \geq 0$. In: The Homology of Iterated Loop Spaces. Lecture Notes in Mathematics, vol 533. Springer, Berlin, Heidelberg 1976 (\href{https://doi.org/10.1007/BFb0080467}{doi:10.1007/BFb0080467})} \end{itemize} \hypertarget{as_chernsimons_amplitudes}{}\subsubsection*{{As Chern-Simons amplitudes}}\label{as_chernsimons_amplitudes} Discussion of higher order Vassiliev invariants as [[Chern-Simons theory]]-[[correlators]], hence as [[configuration space of points|configuration space]]-[[integrals]] of [[wedge products]] of [[Chern-Simons propagators]] assigned to [[edges]] of [[Feynman diagrams]] in the [[graph complex]]: \begin{itemize}% \item \hyperlink{Kontsevich93}{Kontsevich 93, Section 5} \item Daniel Altschuler, Laurent Freidel, \emph{Vassiliev knot invariants and Chern-Simons perturbation theory to all orders}, Commun. Math. Phys. 187 (1997) 261-287 (\href{https://arxiv.org/abs/q-alg/9603010}{arxiv:q-alg/9603010}) \item [[Alberto Cattaneo]], Paolo Cotta-Ramusino, Riccardo Longoni, \emph{Configuration spaces and Vassiliev classes in any dimension}, Algebr. Geom. Topol. 2 (2002) 949-1000 (\href{https://arxiv.org/abs/math/9910139}{arXiv:math/9910139}) \item [[Alberto Cattaneo]], Paolo Cotta-Ramusino, Riccardo Longoni, \emph{Algebraic structures on graph cohomology}, Journal of Knot Theory and Its Ramifications, Vol. 14, No. 5 (2005) 627-640 (\href{https://arxiv.org/abs/math/0307218}{arXiv:math/0307218}) \end{itemize} Reviewed in: \begin{itemize}% \item [[Ismar Volić]], Section 4 of: \emph{Configuration space integrals and the topology of knot and link spaces}, \href{https://fdocuments.co/amp/document/morfismos-vol-17-no-2-2013.html}{Morfismos, Vol 17, no 2, 2013} (\href{https://arxiv.org/abs/1310.7224}{arxiv:1310.7224}) \end{itemize} \hypertarget{vassiliev_invariants_of_braids}{}\subsubsection*{{Vassiliev invariants of braids}}\label{vassiliev_invariants_of_braids} \hypertarget{general_2}{}\paragraph*{{General}}\label{general_2} Vassiliev invariants of [[braid group|braids]] via [[chord diagrams]]: \begin{itemize}% \item [[Dror Bar-Natan]], \emph{Vassiliev and Quantum Invariants of Braids}, Geom. Topol. Monogr. 4 (2002) 143-160 (\href{https://arxiv.org/abs/q-alg/9607001}{arxiv:q-alg/9607001}) \end{itemize} \hypertarget{as_observables_on_fuzzy_spheres}{}\paragraph*{{As observables on fuzzy spheres}}\label{as_observables_on_fuzzy_spheres} Relation of [[Dp-D(p+2)-brane bound states]] (\href{Dp-Dp+2-brane+bound+states#ReferencesRelationToMonopoles}{hence} [[Yang-Mills monopoles]]) to [[Vassiliev braid invariants]] via [[chord diagrams]] computing [[radii]] of [[fuzzy spheres]]: \begin{itemize}% \item [[Sanyaje Ramgoolam]], [[Bill 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 [[Simon McNamara]], [[Constantinos Papageorgakis]], [[Sanyaje Ramgoolam]], [[Bill 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 [[Simon McNamara]], Section 4 of: \emph{Twistor Inspired Methods in Perturbative FieldTheory and Fuzzy Funnels}, 2006 (\href{http://inspirehep.net/record/1351861}{spire:1351861}, \href{https://strings.ph.qmul.ac.uk/sites/default/files/Mcnamaraphd.pdf}{pdf}, [[McNamara06.pdf:file]]) \item [[Constantinos Papageorgakis]], p. 161-162 of: \emph{On matrix D-brane dynamics and fuzzy spheres}, 2006 ([[Papageorgakis06.pdf:file]]) \end{itemize} category: geometry, topology [[!redirects Vassiliev knot invariant]] [[!redirects Vassiliev knot invariants]] [[!redirects Vassiliev invariants]] [[!redirects Vassiliev finite type invariants]] [[!redirects Vassiliev finite type invariant]] [[!redirects Vassiliev finite-type invariants]] [[!redirects Vassiliev finite-type invariant]] [[!redirects finite type invariants]] [[!redirects finite type invariant]] [[!redirects finite-type invariants]] [[!redirects finite-type invariant]] [[!redirects Vassiliev invariants of braids]] [[!redirects Vassiliev braid invariant]] [[!redirects Vassiliev braid invariants]] \end{document}