COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

338 J. H. Conway Enumeration of knots and links 339
K % ,L is a product of K and L, obtained by tying each of them sepa- The third identity involves possibly three distinct string labels. If Kl yields
rately in a string labelled r, then K,, KS, and K4 on the replacement of
VK%,L = VK-{~}.VL.
Our tables list only knots which are prime in the sense of such products,
and the assumption of primality is implicit elsewhere in this paper.
Our potential function is related to the Alexander polynomial il,
by the identity
&(r2) = {I}-V&l
if K is a proper knot, and by
AK(r2, s2, . . .) = V&r,s, . . .)
otherwise, but it is important to realize that A, is defined to within multiplication
by powers of the variables and - 1, while OK is defined abso-
lutely.
The most important and valuable properties of the potential function
are for this reason not shared by the polynomial. Let K0 yield K+ and
K- on replacement of the tangle
ur
‘3 by X1 and ,LPr respectively,
r r r
the labellings and orientations being significant. Then we have
OK+ = VK- + {r}- pk,,
called the first identity, which enables us to compute any one of the three
potentials from the other two.
The second identity relates knots KOO, K++, K--, defined as above, but
now using the tangles
or alternatively
The second identity asserts that
in the first case, and
in the second case.
L-Ar
f &qr and dbr
F-s
s
VK++ + VK-- = {rs}.VKw
VK+++ VL- = {r-ls}~V~o,
then we have
where now the labellings are immaterial.
These identities have many consequences which we cannot explore in
detail here, although we shall give a few examples. Let t be a tangle whose 4
emerging strings are oriented and labelled as in Fig. 12. Define the polynomialfiaction
oft as the formal fraction
{r }- VK
{r}- VL
where K and L are the knots 1 * t and 1 * to. Then the identities which we
asserted for determinant fractions in Section 4 hold also for polynomial
fractions.
lxL c
FIG. 12
If we consider generalized tangles with 2n emerging arcs instead of 4
(such as, for 2n = 6, those of the third identity), then we can determine
the potential of any knot obtained by joining the emergent arcs of two such
tangles in terms of n! potential functions associated with each tangle separately,
provided that all the emerging strings have the same label. In the
case n = 2 the 2! potentials are the numerator and denominator of the
polynomial fraction. It becomes natural to think of such tangles as being
-to within a certain equivalence relation-elements of a certain vector
space in which our identities become linear relations, and there are many
natural questions we can ask about this space. However, when the emerging
arcs may have distinct labels, it is not even known whether the dimension
of the tangle space is finite.
We have not found a satisfactory explanation of these identities, although
we have verified them by reference to a “normalized” form of the ‘L-matrix
definition of the Alexander polynomial, obtained by associating the rows
and columns in a natural way. This normalization is useful in other ways