On the First Two Vassiliev Invariants

Abstract

The values that the first two Vassiliev invariants take on prime knots with up to fourteen crossings are considered. This leads to interesting fish-like graphs. Several results about the values taken on torus knots are proved.'First the fish must be caught.' That is easy: a baby, I think, could have caught it.

Models of random knots

“…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.…”

Problems on invariants of knots and 3–manifolds

“…Remark Willerton [401] observed that the set of (v 2 (K), v 3 (K)) for knots K with a (certain) fixed crossing number gives a fish-like graph. This fish-like graph is discussed in [103] from the point of view of the Jones polynomial.…”

“…Conjecture 2.11 (S. Willerton [401]) Let v 3 be as above. If a knot K has a diagram with n crossings, then…”

Tree invariants and Milnor linking numbers with indeterminacy

“…Theorem of Goussarov [10] shows that any finite type invariant v of knots can be expressed as P v , · for a suitable choice of the arrow polynomial P v . Arrow polynomials of some low degree invariants have been computed in [27,29,30,35]. For instance, the second coefficient of the Conway polynomial c 2 (K) of a knot K, represented by a Gauss diagram G K is given by , G K (c.f.…”

“…35) where α corresponds to e and r(S) + l(R) = j − i − 1. Given I ⊂ [n], such that j ∈ I, J = [n] − I, define A (ιιι.b) = A (ιιι.b) (i, j; I; t) to be the set of trees in A(n; t) which can be decomposed according to(6.35) with I(R; A) = I, I(U ; A) = J and let I (ιιι.b) (i, j) = {I ⊂ [n] | A (ιιι.b) (i, j; I; t) = ∅}.…”