Polynomial invariants and Vassiliev invariants

2002/11/04 by Myeong-Ju Jeong, Chan-Young Park
Mathematics · #math.GT #msc:57M25

paper · pdf

published as Geom. Topol. Monogr. 4 (2002) 89-101 · Published by Geometry and Topology Monographs at http://www.maths.warwick.ac.uk/gt/GTMon4/paper7.abs.html

arxiv created 2002/11/04 · arxiv updated 2009/11/30

Abstract

We give a criterion to detect whether the derivatives of the HOMFLY polynomial at a point is a Vassiliev invariant or not. In particular, for a complex number b we show that the derivative P_K^(m,n)(b,0)=d^m/da^m dⁿ/dxⁿ P_K(a,x)|(a, x) = (b, 0) of the HOMFLY polynomial of a knot K at (b,0) is a Vassiliev invariant if and only if b= -+1. Also we analyze the space V_n of Vassiliev invariants of degree <=n for n = 1,2,3,4,5 by using the bar-operation and the star-operation in [M-J Jeong, C-Y Park, Vassiliev invariants and knot polynomials, to appear in Topology and Its Applications]. These two operations are unified to the hat-operation. For each Vassiliev invariant v of degree <=n, hat(v) is a Vassiliev invariant of degree <=n and the value hat(v)K) of a knot K is a polynomial with multi-variables of degree <=n and we give some questions on polynomial invariants and the Vassiliev invariants.

Related

• Some minimal degree Vassiliev invariants not realizable by the Homfly and Kauffman polynomial
• Polynomial Invariants are Polynomial
• On Finiteness of Certain Vassiliev Invariants
• On a Poincaré polynomial from Khovanov homology and Vassiliev invariants
• Vassiliev Invariants for Torus Knots