arXiv subjects

Carolyn Engelhardt

Publications and source records attributed to Carolyn Engelhardt.

2 recordsLinked to original sources

arXiv2026

Bridge Multisections of Knotted Surfaces in $S^4$

Bridge multisections are combinatorial descriptions of surface links in 4-space using tuples of trivial tangles. They were introduced by Islambouli, Karimi, Lambert-Cole, and Meier to study curves in rational surfaces. In this paper, we prove a uniqueness result for bridge multisections of surfaces in 4-space: we give a complete set of moves relating to any two multiplane diagrams of the same surface. This is done by developing a surgery operation on multiplane diagrams called band surgery. Another application of this surgery move is that any $n$-valent graph with an $n$-edge coloring is the spine of a bridge multisection for an unknotted surface. We also prove that any multisected surface in $S^4$ can be unknotted by finitely many band surgeries.

math.GT↗

arXiv2024

Generating Infinitely Many Hyperbolic Knots with Plats

In this paper we study the relationships between links in plat position, the dynamics of the braid group, and Heegaard splittings of double branched covers of $S^3$ over a link. These relationships offer new ways to view links in plat position and a new tool kit for analyzing links. In particular, we show that the Hempel distance of the Heegaard splitting of the double branched cover obtained from a plat is a lower bound for the Hempel distance of that plat. Using the Hempel distance of a knot in bridge position and pseudo-Anosov braids we obtain our main result: a construction of infinitely many sequences of prime hyperbolic $n$-bridge knots for $n \geq 3$, infinitely many of which are distinct. We consider known results to show that the knot genus and hyperbolic volume of these knots are bounded below by a linear function.

math.GT↗