Sergei Chmutov, Sergei Duzhin, Jacob Mostovoy, Section 4 of: Introduction to Vassiliev knot invariants, Cambridge University Press, 2012 ( arxiv:1103.5628, doi:10.1017/CBO9781139107846)ĭavid Jackson, Iain Moffat, Section 11 of: An Introduction to Quantum and Vassiliev Knot Invariants, Springer 2019 ( doi:10.1007/978-3-3) Original discussion of chord diagrams in the context of Vassiliev invariants:ĭror Bar-Natan, On the Vassiliev knot invariants, Topology Volume 34, Issue 2, April 1995, Pages 423-472 ( doi:10.1016/0040-9383(95)93237-2, pdf)ĭror Bar-Natan, Vassiliev and Quantum Invariants of Braids, Geom. Everett, On a class of linked diagrams II. Horizontal weight systems are cohomology of loop space of configuration spaceĮarly consideration of round chord diagrams (as âlinked diagramsâ): Weight systems are associated graded of Vassiliev invariants However, such chord diagrams nonetheless play an important role in the theory of Vassiliev invariants, by the theorem that the value of a Vassiliev invariant v v of degree ⤠n \le n on any singular knot K K with n n simple double points depends only on the chord diagram c ( K ) c(K). This chord diagram c ( K ) c(K) of course does not faithfully represent the original knot K K (e.g., it does not include any information about over-crossings and under-crossings). Chord diagrams of singular knotsīy an analogous mechanism, any singular knot K K with n n simple double points (i.e., points where the knot intersects itself transversally) gives rise to a chord diagram c ( K ) c(K) with n n chords, by connecting the preimages of these double points. If we would label the points of a chord diagram,ĭ : + ⦠+ â n times â ĭ : \underset^3, but into a thickened surface of arbitrary genus. (For topological definitions of both unrooted and rooted chord diagrams, see for example Chapter 6 of Lando and Zvonkin, or Definition 1.5 of Bar-Natan 1995 for the unrooted case.) A combinatorial definition We present only the combinatorial definition here, starting from rooted chord diagrams and using those to define unrooted chord diagrams, rather than the other way around. DefinitionsĬhord diagrams can be defined both in topological terms, which formalize the graphical intuition, as well as in purely combinatorial terms. Under standard equivalence relations these are actually equivalent to chord diagrams, see below). If also (trivalent) internal vertices are considered, one speaks of Jacobi diagrams. As codes for holographic entanglement entropyĪ chord diagram is a finite trivalent undirected graph with an embedded oriented circle and all vertices on that circle, regarded modulo cyclic identifications, if any.Įquivalently this is a pairing (by chords) of all elements in a cyclic order (the boundary vertices).Ĭhord diagrams are a basic object of study in combinatorics with remarkably many applications in mathematics and physics, notably in knot theory and Chern-Simons theory (where they control Vassiliev invariants) but more recently also in stringy quantum gravity (see the references below).In D p p/D ( p + 2 ) (p+2)-brane intersections.In AdS 2 / CFT 1 AdS_2/CFT_1, JT-gravity/SYK-model.For single trace operators in AdS/CFT duality.Chord diagrams and weight systems in Physics.Gauss diagrams of ordinary (virtual) knots.
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