FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2005, VOLUME 11, NUMBER 4, PAGES 119-126

**Knot theory and the Casson invariant in Artin presentation theory**

J. S. Calcut

Abstract

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In Artin presentation theory, a smooth, compact four-manifold is
determined by a certain type of presentation of the fundamental
group of its boundary.
Topological invariants of both three- and four-manifolds can be
calculated solely in terms of functions of the discrete Artin
presentation.
González-Acuña proposed such a formula for the Rokhlin
invariant of an integral homology three-sphere.
This paper provides a formula for the Casson invariant of
rational homology spheres.
Thus, all 3D Seiberg--Witten invariants can be calculated by using
methods of theory of groups in Artin presentation theory.
The Casson invariant is closely related to canonical knots determined
by an Artin presentation.
It is also shown that any knot in any three-manifold appears as
a canonical knot in Artin presentation theory.
An open problem is to determine 4D Seiberg--Witten and Donaldson
invariants in Artin presentation theory.

Location: http://mech.math.msu.su/~fpm/eng/k05/k054/k05410h.htm

Last modified: November 29, 2005