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In mathematics, the 'Alexander polynomial' is a knot invariant which assigns a polynomial with integer coefficients to each knot type. J. W. Alexander 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.^[1]

Contents

Definition

Computing the polynomial

Basic properties of the polynomial

Geometric significance of the polynomial

Alexander-Conway polynomial

References

External links

Endnotes

Definition

Let ''K'' be a knot in the 3-sphere. Let ''X'' be the infinite cyclic cover of the knot complement of ''K''. There is a covering transformation ''t'' acting on ''X''. Consider the first homology (with integer coefficients) of ''X'', denoted $H\_1(X)$. The transformation ''t'' acts on the homology and so we can consider $H\_1(X)$ a module over $mathbb\{Z\}[t,\; t^\{-1\}]$. This is called the 'Alexander invariant'.
The module if finitely presentable; a presentation matrix for this module is called the 'Alexander matrix'. If the number of generators, ''r'', is less than or equal to the number of relations, ''s'', then we consider the ideal generated by all ''r'' by ''r'' minors of the matrix; this is the 'Alexander ideal' and does not depend on choice of presentation matrix. If ''r > s'', set the ideal equal to 0. If the Alexander ideal is principal, take a generator; this is called a Alexander polynomial of the knot. Since this is only unique up to multiplication by the Laurent monomial $pm\; t^n$, one often fixes a particular unique form. Alexander's choice of normalization is to make the polynomial have a positive constant term.
Alexander proved that the Alexander ideal is nonzero and always principal. Thus an Alexander polynomial always exists, and is clearly a knot invariant, denoted $Delta\_K(t)$.

Computing the polynomial

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'' crossings; there are ''n'' + 2 regions of the knot diagram. To work out the Alexander polynomial, firstly one must create an incidence matrix of size (''n'', ''n'' + 2). The ''n'' rows correspond to the ''n'' crossings, and the ''n'' + 2 columns to the regions. The values for the matrix entries are either 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'' by ''n'' matrix. Depending on the columns removed, the answer will differ by multiplication by $pm\; t^n$. To resolve this ambiguity, divide out the largest possible power of ''t'' and multiply by -1 if necessary, so that the constant term is positive. This gives the Alexander polynomial.
The Alexander polynomial can also be computed from the Seifert matrix.

Basic properties of the polynomial

The Alexander polynomial is symmetric: $Delta\_K(t^\{-1\})\; =\; Delta\_K(t)$
and it evaluates to a unit on 1: $Delta\_K(1)=pm\; 1$.
It is known that every integral Laurent polynomial which is both symmetric and evaluates to a unit at 1 is the Alexander polynomial of a knot (Kawauchi 1996).

Geometric significance of the polynomial

Since the Alexander ideal is principal, $Delta\_K(t)=1$ if and only if the commutator subgroup of the knot group is perfect.
Consider the 3-sphere to be the boundary of the 4-ball. Michael Freedman proved that a knot in the 3-sphere is topologically slice, i.e. bounds a tame topological disc in the 4-ball, if and only if the Alexander polynomial of the knot is trivial (Freedman and Quinn, 1990)
Knots with symmetries are known to have restricted Alexander polynomials. See the symmetry section in (Kawauchi 1996). Although, 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'' (highest and lowest order terms equal to $pm\; 1$). In fact, if $S\; o\; C\_K\; o\; S^1$ is a fiber bundle where $C\_K$ is the knot complement, let $g\; :\; S\; o\; S$ represent the monodromy, then $Delta\_K(t)\; =\; Det(tI-g\_$
★ ) where $g\_$
★ : H_1 S o H_1 S is the induced map on homology.
==Relations to satellite operations==
If a knot $K$ is a satellite knot with companion $K\text{'}$ ie: there exists an embedding $f\; :\; S^1\; imes\; D^2\; o\; S^3$ such that $K=f(K\text{'})$ where $S^1\; imes\; D^2\; subset\; S^3$ is an unknotted solid torus, then $Delta\_K(t)\; =\; Delta\_\{f(S^1\; imes\; \{0\}\}(t)\; Delta\_\{K\text{'}\}(t^a)$. Where $a\; in\; mathbb\; Z$ is the integer that represents $K\text{'}\; subset\; S^1\; imes\; D^2$ in $H\_1(S^1\; imes\; D^2)\; =\; mathbb\; Z$.
Examples: For a connect-sum $Delta\_\{K\_1\; \#\; K\_2\}(t)\; =\; Delta\_\{K\_1\}(t)\; Delta\_\{K\_2\}(t)$. If $K$ is an untwisted Whitehead double, then $Delta\_K(t)=pm\; 1$.

Alexander-Conway polynomial

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 $$
abla(z) and called the 'Alexander-Conway polynomial' (also known as 'Conway polynomial' or 'Conway-Alexander polynomial').
Suppose we are given an oriented link diagram, where $L\_+,\; L\_-,\; L\_0$ are link diagrams resulting crossing and smoothing changes on a local region of a specified crossing of the diagram, as indicated in the figure.
Here are Conway's skein relations:

★ $$
abla(O) = 1 (where O is any diagram of the unknot)

★ $$
abla(L_+) -
abla(L_-) = z
abla(L_0)
The relationship to the standard Alexander polynomial is given by $Delta\_L(t^2)\; =$
abla_L(t - t^{-1}). Here $Delta\_L$ must be properly normalized (by multiplication of $pm\; t^\{n/2\}$) to satisfy the skein relation $Delta(L\_+)\; -\; Delta(L\_-)\; =\; (t^\{1/2\}\; -\; t^\{-1/2\})\; Delta(L\_0)$. Note that this relation gives a Laurent polynomial in ''t^1/2''.
See knot theory for an example computing the Conway polynomial of the trefoil.

References

★ J.W. Alexander, ''Topological invariants of knots and links''. Trans. Amer. Math. Soc. 30 (1928), no. 2, 275--306.[1]

★ Colin C. Adams, ''The Knot Book: An elementary introduction to the mathematical theory of knots.'' Revised reprint of the 1994 original. American Mathematical Society, Providence, RI, 2004. xiv+307 pp. ISBN 0-8218-3678-1 (accessible introduction utilizing a skein relation approach)

★ Michael H. Freedman and Frank Quinn, Topology of 4-manifolds, Princeton Mathematical Series, vol 39, Princeton University Press, Princeton, NJ, 1990. ISBN 0-691-08577-3

★ Akio Kawauchi, ''A Survey of Knot Theory'', Birkhauser 1996 (covers several different approaches, explains relations between different versions of the Alexander polynomial)

★ Dale Rolfsen, ''Knots and Links'', 2nd edition, Publish or Perish, 1990 ISBN 0-914098-16-0 (explains classical approach using the Alexander invariant; knot and link table with Alexander polynomials)

External links

★ Knot Atlas - knot and link tables with computed Alexander and Conway polynomials

Endnotes

1. Alexander describes his skein relation toward the end of his paper under the heading "miscellaneous theorems", which is possibly why it got lost. Joan Birman mentions in her paper ''New points of view in knot theory'' (Bull. Amer. Math. Soc. (N.S.) 28 (1993), no. 2, 253--287) that Mark Kidwell brought her attention to Alexander's relation in 1970.