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