In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923. In 1969, John Conway showed a version of this polynomial, now called the Alexander–Conway polynomial, could be computed using a skein relation, although its significance was not realized until the discovery of the Jones polynomial in 1984. Soon after Conway's reworking of the Alexander polynomial, it was realized that a similar skein relation was exhibited in Alexander's paper on his polynomial.
Let K be a knot in the 3-sphere. Let X be the infinite cyclic cover of the knot complement of K. This covering can be obtained by cutting the knot complement along a Seifert surface of K and gluing together infinitely many copies of the resulting manifold with boundary in a cyclic manner. There is a covering transformation t acting on X. Consider the first homology (with integer coefficients) of X, denoted
H1(X)
H1(X)
Z[t,t-1]
The module is finitely presentable; a presentation matrix for this module is called the Alexander matrix. If the number of generators,
r
s
r x r
r>s
\pmtn
Alexander proved that the Alexander ideal is nonzero and always principal. Thus an Alexander polynomial always exists, and is clearly a knot invariant, denoted
\DeltaK(t)
The following procedure for computing the Alexander polynomial was given by J. W. Alexander in his paper.
Take an oriented diagram of the knot with
n
n+2
(n,n+2)
n
n
n+2
0,1,-1,t,-t
Consider the entry corresponding to a particular region and crossing. If the region is not adjacent to the crossing, the entry is 0. If the region is adjacent to the crossing, the entry depends on its location. The following table gives the entry, determined by the location of the region at the crossing from the perspective of the incoming undercrossing line.
on the left before undercrossing:
-t
on the right before undercrossing:
1
on the left after undercrossing:
t
on the right after undercrossing:
-1
Remove two columns corresponding to adjacent regions from the matrix, and work out the determinant of the new
n x n
\pmtn
n
t
-1
The Alexander polynomial can also be computed from the Seifert matrix.
After the work of J. W. Alexander, Ralph Fox considered a copresentation of the knot group
3\backslash | |
\pi | |
1(S |
K)
\DeltaK(t)
The Alexander polynomial is symmetric:
-1 | |
\Delta | |
K(t |
)=\DeltaK(t)
\overline{H1X}\simeq
Hom | |
Z[t,t-1] |
(H1X,G)
G
Z[t,t-1]
Z[t,t-1]
Z[t,t-1]
\overline{H1X}
Z[t,t-1]
H1X
H1X
t
t-1
Furthermore, the Alexander polynomial evaluates to a unit on 1:
\DeltaK(1)=\pm1
From the point of view of the definition, this is an expression of the fact that the knot complement is a homology circle, generated by the covering transformation
t
M
rank(H1M)=1
\DeltaM(t)
\DeltaM(1)
H1M
Every integral Laurent polynomial which is both symmetric and evaluates to a unit at 1 is the Alexander polynomial of a knot.[1]
Since the Alexander ideal is principal,
\DeltaK(t)=1
For a topologically slice knot, the Alexander polynomial satisfies the Fox–Milnor condition
\DeltaK(t)=f(t)f(t-1)
f(t)
Twice the knot genus is bounded below by the degree of the Alexander polynomial.
Michael Freedman proved that a knot in the 3-sphere is topologically slice; i.e., bounds a "locally-flat" topological disc in the 4-ball, if the Alexander polynomial of the knot is trivial.
Kauffman describes the first construction of the Alexander polynomial via state sums derived from physical models. A survey of these topic and other connections with physics are given in.
There are other relations with surfaces and smooth 4-dimensional topology. For example, under certain assumptions, there is a way of modifying a smooth 4-manifold by performing a surgery that consists of removing a neighborhood of a two-dimensional torus and replacing it with a knot complement crossed with S1. The result is a smooth 4-manifold homeomorphic to the original, though now the Seiberg–Witten invariant has been modified by multiplication with the Alexander polynomial of the knot.
Knots with symmetries are known to have restricted Alexander polynomials. Nonetheless, the Alexander polynomial can fail to detect some symmetries, such as strong invertibility.
If the knot complement fibers over the circle, then the Alexander polynomial of the knot is known to be monic (the coefficients of the highest and lowest order terms are equal to
\pm1
S\toCK\toS1
CK
g:S\toS
\DeltaK(t)={\rmDet}(tI-g*)
g*\colonH1S\toH1S
If a knot
K
K'
f:S1 x D2\toS3
K=f(K')
S1 x D2\subsetS3
K'
\DeltaK(t)=
\Delta | |
f(S1 x \{0\ |
)}(ta)\DeltaK'(t)
a\inZ
K'\subsetS1 x D2
1 x | |
H | |
1(S |
D2)=Z
Examples: For a connect-sum
\Delta | |
K1\#K2 |
(t)=
\Delta | |
K1 |
(t)
\Delta | |
K2 |
(t)
K
\DeltaK(t)=\pm1
Alexander proved the Alexander polynomial satisfies a skein relation. John Conway later rediscovered this in a different form and showed that the skein relation together with a choice of value on the unknot was enough to determine the polynomial. Conway's version is a polynomial in z with integer coefficients, denoted
\nabla(z)
Suppose we are given an oriented link diagram, where
L+,L-,L0
Here are Conway's skein relations:
\nabla(O)=1
\nabla(L+)-\nabla(L-)=z\nabla(L0)
The relationship to the standard Alexander polynomial is given by
2) | |
\Delta | |
L(t |
=\nablaL(t-t-1)
\DeltaL
\pmtn/2
\Delta(L+)-\Delta(L-)=(t1/2-t-1/2)\Delta(L0)
See knot theory for an example computing the Conway polynomial of the trefoil.
Using pseudo-holomorphic curves, Ozsváth-Szabó and Rasmussen associated a bigraded abelian group, called knot Floer homology, to each isotopy class of knots. The graded Euler characteristic of knot Floer homology is the Alexander polynomial. While the Alexander polynomial gives a lower bound on the genus of a knot, showed that knot Floer homology detects the genus. Similarly, while the Alexander polynomial gives an obstruction to a knot complement fibering over the circle, showed that knot Floer homology completely determines when a knot complement fibers over the circle. The knot Floer homology groups are part of the Heegaard Floer homology family of invariants; see Floer homology for further discussion.