Geometry & Topology Monographs, Vol. 4 (2002),
Invariants of knots and 3-manifolds (Kyoto 2001),
Paper no. 7, pages 89--101.
Polynomial invariants and Vassiliev invariants
Myeong-Ju Jeong, Chan-Young Park
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 d^n/dx^n 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.
Keywords.
Knots, Vassiliev invariants, double dating tangles, knot polynomials
AMS subject classification.
Primary: 57M25.
E-print: arXiv:math.GT/0211045
Submitted to GT on 29 November 2001.
(Revised 7 March 2002.)
Paper accepted 22 July 2002.
Paper published 28 July 2002.
Notes on file formats
Myeong-Ju Jeong, Chan-Young Park
Department of Mathematics, College of Natural Sciences
Kyungpook National University, Taegu 702-701 Korea
Email: determiner@hanmail.net, chnypark@knu.ac.kr
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