COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

340 J. H. Conway
-thus our symmetry formulae show that a 2-component link can only be
amphicheiral if its polynomial vanishes identically. It seems plain -that
much work remains to be done in this field.
7. Determinants and signature. We define the reduced polynomial DK(x)
by the equation
DK(X) = {x}. V&x, x, . . .),
and the determinant S = 6, as the number a&i). Our determinant differs
from the usual one only by a power of i. The potential identities of the last
section yield determinant identities when we put i for each variable.
Murasugi [7] has defined invariants called the signature, UK, and nullity,
nK, and described some of their properties. These invariants depend on the
string orientations of K, but not on its labelling. We shall describe enough
of their properties to enable their calculation to proceed in much the same
way as that of the potential function.
For any knot K we have the identity
where 6% = /&I, and the condition
ItK =- 1 if and only if aK = 0,
the first of which determines UK modulo 4 provided gK $ 0. But one of
Murasugi’s results is that
whenever K+ and KO are related as in the first identity. These two results
determine 0, completely in almost all cases, and make its calculation very
swift indeed. Of course it should be remembered that a, and nK are integers,
and 1 4 izK=% IKI.
We have the relations
concerning obverses and products, and if we define a$ as oK - A,, where &
is the total linking of K (the sum of the linking numbers of each pair of
distinct strings of K), then the reorienting identity is that &, like Sg, is an
invariant of the unoriented knot K. In the tables, we give only these resid&
invariants, 60 being the numerator of the rational fraction which we give
for rational knots.
8. Slice knots and the cobordism group. A proper knot which can arise
as the central 3-dimensional section of a (possibly knotted) locally flat
2-sphere in 4-space we call a slice knot. A natural application of our tables
is to the discovery of interesting slice knots, since for slice knots there are
simple conditions on the polynomial, signature, and Minkowski units. In
Enumeration of knots and links 341
particular, we might hope to find a slice knot which is not a ribbon knot
[3], since several published proofs that all slice knots are ribbon have been
found to be fallacious.
The slice knots with 10 crossings or less were found to be
-,42, 3113,2312,(3,21,2-),212112,20:20:20,(3,3,21-),
64, 3313,2422,2211112,(41, 3,2),(21,21, 21+);22-20,
a2.2.2 0.2 0, 1 0*, (3 2, 2 1, 2-j, (2 2, 2 1 1, 2-), (4, 3, 2 l-),
((3, 2)-(2 1, 2)), 3 : 2 : 2,
together with the composite knots 3 % 3,2 2 EX 2 2,5 % 5,3 2 % 3 2. The granny
knot 3 % 3 and the knot a2.2.2.2 0 satisfy the polynomial condition but
not the signature condition, and so are not slice knots.
Now suppose that in the slab of 4-space bounded by two parallel 3-spaces
we have an annulus (PXZ) whose boundary circles lie in the two 3-spaces.
Then the two knots defined by these circles we call cobordant. Cobordism is
an equivalence relation, and the cobordism classes form a group under the
product operation, the unit class being the class of slice knots, and the inverse
of any class is the class of inverse knots to the knots of that class.
A search was made for cobordances between knots of at most 10 crossings
and knots of at most 6 crossings, which in addition to the cobordances
between slice knots, found only
3~321223, 21, 2~222112~22, 211, 2~31, 3, 21~211, 3, 21-a
22=3,21,21~3,21,2+
5r3412r4,3,21
32~31,3,21~3,3,21+r3,21,2++
3 % 3 “= .2.2.2.2 0,
all of these being to within sign and orientation.
All slice knots given were found to be ribbon knots. However, thepresence
of the particular knot lO* = (2 X 5)* leads us to examine the more
general a strand b bight Turk’s Head knot ((a - 1) x b)*. Andrew Tristram
has proved that if a and b are odd and coprime this knot obeys all known
algebraic conditions for slice knots, but despite a prolonged attack the only
cases definitely known to be slice are the trivial cases with a or b = 1, and
the cases a = 3, b = 5, and a = 5, b = 3. Since most of our methods for
proving knots slice would also prove them ribbon, the way is left open for
a conjecture that some of least of these are slice knots which are not ribbon
knots.
9. Notes on the tables. Acknowledgments. The tables (pp. 343-357) list
all proper knots of at most 11 crossings, and all links of at most 10 crossings,
with various invariants tabulated over parts of this range. Knots
listed separately are believed to be distinct, and the symmetries listed
under S are believed to be a complete set. (The evidence is very strong-