arXiv subjects
Publications and source records attributed to Corbin Reid.
\emph{Magic braids} are used in leatherworking to make intricate straps and bands. They appear to be impossible to make. To determine which braids can be made with this leatherworking technique, we explore their relation to several braid groups.
We consider volumes of two families of links that have been the focus of recent results on geometry, namely weakly generalised alternating (WGA) links and fully augmented links (FAL). Both have known lower bounds on hyperbolic volume in terms of their diagram combinatorics, but less is known about upper bounds. In fact, Kalfagianni and Purcell recently found a family of WGA knots on a compressible surface for which there can be no upper bounds on volume in terms of twist number. They asked if upper volume bounds always exist on incompressible surfaces. We show the answer is no: we find infinite families of WGA and FALs on incompressible surfaces with no upper bound on volume in terms of twist number.
Consider a robot that remembers only the starting position and walks along a knot once on a knot diagram, switching every undercrossing it meets until it returns to the starting position. We observe that the robot produces an ascending diagram, and we provide a new combinatorial proof that every ascending or descending knot diagram can be transformed into the zero-crossing unknot diagram. Using the machinery developed from the combinatorial proof, we show that the minimal number of Reidemeister moves required for such a transformation is bounded above by (7C+1)C if the diagram has C crossings. Moreover, we provide a new alternative proof that there exist sequences of Reidemeister moves that do not increase the number of crossings and transform ascending or descending knot diagrams into zero-crossing unknot diagrams.
Classical fully augmented links have explicit hyperbolic geometry, and have diagrams on the 2-sphere in the 3-sphere. We generalise to construct fully augmented links projected to the reflection surface of any 3-manifold obtained by doubling a compact 3-manifold and show that the results of the classical setting extend to these links. When the resulting manifolds are hyperbolic, we find bounds on their cusp shapes and volumes. Note these links include virtual fully augmented links, and thus our bounds apply to such links when they are hyperbolic.