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Alexander polynomial

Contents

Idea

In 1928, J. W. Alexander published a paper “Topological Invariants of Knots and Links” in which he defined a polynomial invariant of knots and developed new insights including the braid relations. There are several ways to look at these invariants, some of these use the knot group previously defined by Max Dehn, but there are also various combinatorial methods derived from Alexander’s original one. One of the best known methods is via Fox derivatives and is described in the classical text by Richard Crowell and Ralph Fox.

(…)

Consider some 3-manifold given as a surface fiber bundle over the circle (notice the virtually fibered conjecture). For a fiber surface TT, the translation of the fibre around the base-space circle determines an element in the mapping-class group of TT, a homeomorphism h:TTh\colon T \to T well defined up to isotopy; this element is called the holonomy of the fiber surface; the Alexander polynomial is the characteristic polynomial of the map the holonomy induces on H 1(T)H_1(T).

(Stallings 87)

Analogy with number theory

See Sikora 01, analogy 2.2 (10)) for the comparison in arithmetic topology, where Alexander-Fox theory is the analog of Iwasawa theory (Morishita, section 7).

In Remark 3.3 of Sugiyama 04, the Alexander polynomial is described as the L-function of the knot complement, taken there with the trivial represenation. As such it resembles the local zeta function of a curve.

References

Various approaches to the Alexander polynomial are described in introductory texts such as

An analogue in number theory is Iwasawa polynomial. Cf. for number theoretic analogies

Other works

category: knot theory