UC Davis

The Kakimizu complexes have been studied for various classes of links. O.Kakimizu initially found theKakimizu complexes for knots with crossing numbers less than or equal to 10. Hatcher and Thurston found the 0-skeleton of the Kakimizu complexes of 2-bridge links, while Sakuma later generalized this finding for special arborescent links, describing the Kakimizu complexes for the same. Banks provided a comprehensive proof of results previously announced by Hirasawa and Sakuma, explicitly describing the Kakimizu complexes of non-split, prime special alternating links.

It is established that the Kakimizu complexes of prime, non-split alternating links contain a finite numberof vertices. In this dissertation, we compute the Kakimizu complexes for all 11-crossing prime alternating knots, explicitly describing each and primarily using the methods described above. Some remaining Kakimizu complexes for 11-crossing knots were then determined using Murasugi sums and the sutured manifold theory developed by Gabai, Scharlemann, Kakimizu, and others. Additionally, we apply these computational techniques to the first 1000 knots with 12-crossings, discussing potential obstructions to existing methodologies.