Summary: Comment.Math. Helvetici61 (1986)349-359 0010-2571/86/030349-11501.50+ 0.20/0
Some evaluations of link polynomials
W. B. R. LIC~ORISn and K. C. MILLETr~
For every oriented link L in the 3-sphere there is a 2-variable Laurent
polynomial PL(f, m)~ 7/[/?±1, m±l]. It is defined uniquely by the formulae
(i) Pv = 1 for the unknot U;
(ii) ~?PL++ e-XPL_ + mPL0 =0, where L+, L_, and L0 are any three links
identical except within a ball where they are as shown in Figure I. Details are
given in [F-Y-H-L-M-O] and [L-M 1].
This two-variable polynomial is related to At., the Alexander polynomial, and
Vz., the Jones polynomial, by
PL(i, i(t m -- t-'/Z)) = AL(t),
PL(it -1, --i(t xn -- t- t/Z)) = VL(t).
The purpose of this paper is to evaluate PL for various specific values of (f, m),
giving where possible the interpretation for VL. The values chosen are such that
PL has an elementary form in terms of other known invariants of the link.
Throughout, c(L) denotes the number of components of L.
A few relevant elementary results that can be found in [J] or [L-M 11 are: