nLab Alexander polynomial

Idea

The Alexander polynomial (Alexander 1928) is a polynomial invariant related to braid representation-theory (cf. the Burau representation).

There are several ways to look at thid invariant, 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 $T$, the translation of the fibre around the base-space circle determines an element in the mapping-class group of $T$, a homeomorphism $h\colon T \to T$ 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 $H_1(T)$.

(Stallings 87)

Properties

Analogue in 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.

Related concepts

• Burau representation
• Fox derivative

References

The original article:

• James W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc. 30 (1928) 275–306 $[$doi:10.1090/S0002-9947-1928-1501429-1$]$

Textbook accounts:

• R. H. Crowell and R. H. Fox, Introduction to Knot Theory Springer, Graduate Texts 57 (1963)

• Nick Gilbert, Tim Porter, Knots and surfaces, Oxford University Press (1994) [ISBN:9780198514909]

See also:

• John Stallings, Constructions of fibered knots and links, Proceedings of Symposia in Pure Mathematics 32 (1987) $[$pdf$]$

• Wikipedia, Alexander polynomial

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

• Barry Mazur, Remarks on the Alexander polynomial, pdf

• Masanori Morishita, Analogies between knots and primes, 3-manifolds and number rings, (arXiv:0904.3399)

• Masanori Morishita, Knots and primes: an introduction to arithmetic topology, Springer 2012, chapter 12

• Ken-ichi Sugiyama, The properties of an L-function from a geometric point of view, 2007 pdf

• Ken-ichi Sugiyama, A topological $\mathrm{L}$ -function for a threefold, 2004 pdf;

• Ken-ichi Sugiyama An analog of the Iwasawa conjecture for a compact hyperbolic threefold, math.GT/0606010

• Adam S. Sikora, Analogies between group actions on 3-manifolds and number fields, arXiv:0107210

Other works

• Takefumi Nosaka, Twisted cohomology pairings of knots I; diagrammatic computation, arXiv:1602.01129; Twisted cohomology pairings of knots II; to classical invariants, arXiv:1602.01131

• V. Mishnyakov, A. Sleptsov, N. Tselousov, A new symmetry of the colored Alexander polynomial, Ann. Henri Poincaré 22 (2021) 1235–1265 (doi arXiv:2001.10596)

category: knot theory