arXiv subjects

Seth Hovland

Publications and source records attributed to Seth Hovland.

5 recordsLinked to original sources

arXiv2026

Every Plat Presentation Admits a Positive Bounded Braid Representative

Classifying link types via their n-bridge positions requires navigating vast Hilden double coset classes H2n\B2n/H2n. We provide a restriction on this by proving that every Hilden double coset admits a positive braid representative whose Dehornoy minimum is explicitly bounded. The proof relies on the key observation that $Δ\in H_{2n}$, which allows us to utilize the Garside left-greedy normal form to remove all powers of $Δ$ from any given braid representative. We provide an explicit upper bound on its Dehornoy minimum in terms of the Garside length of the initial presentation. This bounded representative restricts the algebraic space of plat presentations, providing a potential computational tool for studying bridge isotopy classes and offering a constructive step toward algebraic approaches to the Bridge Finiteness Conjecture.

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arXiv2026

Bridge Positions and Plat Presentations of Links

This paper investigates the relationship between links in bridge position and plat presentations. We prove that Hilden double coset classes of plat presentations correspond exactly to bridge positions of a link up to bridge isotopy. This correspondence allows us to translate algebraic questions about plat presentations into geometric questions about bridge positions. Using this perspective, we re-establish several results from both frameworks. In particular, we show that there is a unique Hilden double coset class for the $n$-bridge unknot in $S^3$, and that torus knots admit a single double coset class in plat position. Finally, we discuss how this correspondence can be used to study plat closures of braids, which is the subject of ongoing research.

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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.

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arXiv2024

The Hilden Double Coset Problem in Braid Groups

In this paper we provide a solution to the double coset problem for the braid group $B_n$ modulo the Hilden subgroup $H_n.$ This result demonstrates that, as in the case of braid closures, the Link Problem for plat closures is "stably equivalent" to a solvable algebraic problem. A particularly interesting feature of the proof is that, like Garside's solutions to the Word and Conjugacy Problems, it too relies on Garside's decomposition of braids in $B_n.$

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arXiv2023

The Restriction of Efficient Geodesics to the Non-separating Complex of Curves

In the complex of curves of a closed orientable surface of genus $g,$ $\mathcal{C}(S_g),$ a preferred finite set of geodesics between any two vertices, called \emph{efficient geodesics} introduced by Birman, Margalit, and Menasco in \cite{birman_margalit_menasco_2016}. The main tool used to establish the existence of efficient geodesics was a \emph{dot graph}, which recorded the intersection pattern of a reference arc with the simple closed curves associated with a geodesic path. The idea behind the construction was that a geodesic that is not initially efficient contains shapes in its corresponding dot graph. These shapes then correspond to surgeries that reduce the intersection with the reference arc. In this paper, we show that the efficient geodesic algorithm is able to be restricted to the non-separating curve complex; the proof of this will involve analysis of the dot graph and its corresponding surgeries. Moreover, we demonstrate that given any geodesic in the complex of curves we may obtain an efficient geodesic whose vertices, with the possible exception of the endpoints, are all nonseparating curves.

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