Participants
Heather McGregor
Symmetric unions are structures within knot theory constructed by taking the union of a knot and its mirror image. A knot invariant is a structure which is unchanged when Reidemeister moves are applied to its diagram. This project focused on the ‘symmetric’ Reidemeister moves coined by Eisermann and Lamm, and their on and off-axis variations.
Drawing on previous research by Eisermann and Lamm, Louis Kauffman, and Morwen Thistlethwaite, this project aimed to examine possibilities of invariants on the symmetric union. The bulk of the research was carried out by examining patterns that emerge when deconstructing a knot to its set of spanning trees. Candidates for invariance were, for example, the number of symmetric spanning trees a particular knot had and the sum of signs of edges both on and off-axis.
Research to identify an invariant will continue with an examination of Betti numbers and quotients of spanning trees.