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Polynomial invariants and Vassiliev invariants

Myeong-Ju Jeong and Chan-Young Park

Geometry & Topology Monographs 4 (2002) 89–101

DOI: 10.2140/gtm.2002.4.89

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 PK(m,n)(b,0)=(∂m/∂am)(∂n/∂xn)PK(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 Vn of Vassiliev invariants of degree ≤n for n = 1, 2, 3, 4, 5 by using the ¯ –operation and the *–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 ^–operation. For each Vassiliev invariant v of degree ≤n, v^ is a Vassiliev invariant of degree ≤n and the value 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

Mathematical Subject Classification

Primary: 57M25

References
Publication

Received: 29 November 2001
Revised: 7 March 2002
Accepted: 22 July 2002
Published: 19 September 2002

Authors
Myeong-Ju Jeong
Chan-Young Park
Department of Mathematics
College of Natural Sciences
Kyungpook National University
Taegu 702-701
Korea