Open Access Senior Thesis
Bachelor of Science
© 2020 Forest D Kobayashi
The new work in this document can be broken down into two main parts. In the first, we introduce a formalism for viewing the signed Gauss code for virtual knots in terms of an action of the symmetric group on a countable set. This is achieved by creating a "standard unknot" whose diagram contains countably-many crossings, and then representing tame knots in terms of the action of permutations with finite support. We present some preliminary computational results regarding the group operation given by this encoding, but do not explore it in detail. To make the encoding above formal, we require the aforementioned "unknot with a countable sequence of crossings;" building up the machinery to work with these kinds of objects is the focus of the second part of the project. Initially, the presence of infinitely-many crossing might appear to be a contradiction to the finiteness constraint in Reidemeister's theorem; we show that this is not the case, and introduce the notion of feral points to represent areas of our diagrams in which it is not immediately obvious whether the knot is wild or tame. We employ uniform convergence to create sufficient conditions for guaranteeing ambient isotopy under limits and resolve a seeming contradiction given by the wild arc of Fox-Artin. Finally, we show that any knot whose crossings are topologically discrete is ambient isotopic to a countable union of polygonal segments, and discuss implications for extending Reidemeister's theorem in this context.
Kobayashi, Forest, "Where the Wild Knots Are" (2020). HMC Senior Theses. 234.