On the cohomology theory of knot groups

Abstract

This thesis deals with the cohomology theory of groups, attempting, in particular to find a proof of the following conjecture. Conjecture: All knot groups have cohomological dimension 2. An introduction is given to the theory of knots and knot groups, together with the cohomology theory of a group. A proof of the conjecture for all knot groups has not been found. However, using known results from the theory of groups in general and knot groups in particular, the following classes of knot groups are shown to satisfy the conjecture. a) The class of knot groups which have a presentation involving only one relator. b) The class of groups of alternating knots. c) The class of groups of non-alternating knots with less than ten (10) crossings. d) The class of knot groups whose commutator subgroup is free. A method developed by myself, under the instigation of my supervisor Dr. I.M.S. Dey, shows a necessary and sufficient condition for a knot group to satisfy the conjecture. Using the condition, the following knots are shown to have groups with cohomological dimension 2, provided the assumption made in section 4.6 is correct. e) The group of the Kinoshita Terasaka knot. f) The groups of knots as shown in Fig.l below. g) The groups of knots as shown in Fig. 2 below. A proof for the assumption referred to above has not yet been found. The assumption seems plausible, and can probably be proved using advanced techniques of algebraic topology, with which the author is not familiar at this time.