Systematically Detecting Flypes and Hexagonal Mosaics (Hugh Howards, Wake Forest University)

We talk about building knots using mosaics which were as introduced as a way of modeling quantum knots by Lomonaco and Kauffman and a newer variant, hexagonal mosaics, introduced by Jennifer McLoud-Mann. In the process we find a new bound on crossing numbers for hexagonal mosaics and find an infinite family of knots which do not achieve their hexagonal mosaic number while also in a projection which achieves their crossing number, extending a result of Lew Ludwig et al. In the process we introduce a new tool which makes it easier to systematically recognize when two knots differ by a sequence of Flypes (for example, giving a process to recognize that the Perko Pair were in fact the same knot). No background with mosaics or flypes is necessary. This is joint work with Jiong Li* and Xiotian Liu* (* indicates undergraduate students).