Introduction to Vassiliev Knot Invariants

Abstract

With hundreds of worked examples, exercises and illustrations, this detailed exposition of the theory of Vassiliev knot invariants opens the field to students with little or no knowledge in this area. It also serves as a guide to more advanced material. The book begins with a basic and informal introduction to knot theory, giving many examples of knot invariants before the class of Vassiliev invariants is introduced. This is followed by a detailed study of the algebras of Jacobi diagrams and 3-graphs, and the construction of functions on these algebras via Lie algebras. The authors then describe two constructions of a universal invariant with values in the algebra of Jacobi diagrams: via iterated integrals and via the Drinfeld associator, and extend the theory to framed knots. Various other topics are then discussed, such as Gauss diagram formulae, before the book ends with Vassiliev's original construction.

Models of random knots

“…It is conjectured that knots can be fully classified by Vassiliev's finite type invariants (Vassiliev 1990;Bar-Natan 1995b;Chmutov et al 2012). See Sect.…”

“…Hardness results are known for the knot genus (Agol et al 2006;Lackenby 2016) and for the Jones polynomial (Jaeger et al 1990;Aharonov et al 2009;Kuperberg 2009). Other invariants such as the Alexander-Conway polynomial and finite type invariants are computable in polynomial time (Alexander 1928;Bar-Natan 1995a;Chmutov et al 2012). Many such algorithms are implemented in software packages, such as SnapPy (Culler et al 2016), KnotTheory (Bar-Natan et al 2016b) and KnotScape (Hoste and Thistlethwaite 2016).…”

“…The extension of v to singular knots is given by vðKÞ ¼ vðK This condition is satisfied by several well-studied knot invariants, such as coefficients of knot polynomials (Bar-Natan 1995a; Chmutov et al 2012) and the Kontsevich integral (Bar-Natan 1995b;Chmutov and Duzhin 2001). There is only one knot invariant of order two, up to affine equivalence-the Casson invariant c 2 ðKÞ, which is the coefficient of x 2 in the Alexander-Conway polynomial C K ðxÞ.…”

“…The planar representation of these two invariants follows previous work by Willerton (2002);Chmutov et al (2012) and Ohtsuki et al (2002), who generated scatter plots of ðc 2 ; v 3 Þ for all prime knots with up to n crossings. They similarly obtained fish-shaped figures, although it is unclear how these should scale as the crossing number grows.…”

Three-dimensional extensions of the Alday-Gaiotto-Tachikawa relation

“…The quantities α m,n (K) are the primary Vassiliev invariants [35], which are rational(!) numbers, naturally represented either as modifications of the Gauss linking integrals in the Lorentz gauge ∂ µ A µ = 0 [36], or as the Kontsevich integrals [32] in the holomorphic gauge Az = 0, or through the writhe numbers [37] in the temporal gauge A 0 = 0.…”