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The stick number and the edge length of a knot type in the simple hexagonal lattice (sh-lattice) are the minimal numbers of sticks and edges required, respectively, to construct a knot of the given type in sh-lattice. By introducing a linear transformation between lattices, we prove that for any given knot both values in the sh-lattice are strictly less than the values in the cubic lattice. Finally, we show that the only non-trivial 11-stick knots in the sh-lattice are the trefoil knot ($3_1$) and the figure-eight knot ($4_1$)., Comment: 21 pages, 13 figures

The first two authors introduced vertex distortion and showed that the vertex distortion of the unknot is trivial. It was conjectured that the vertex distortion of a knot is trivial if and only if the knot is trivial. We will use Denne-Sullivan's bound on Gromov distortion to bound the vertex distortion of nontrivial lattice knots. We will then conclude that trivial vertex distortion implies the unknot, proving the conjecture. Additionally, the first conjecture in vertex distortion's debut article is proven and a vertex distortion calculator is given., Comment: 12 pages, 6 figures

We introduce the Geography and Election Outcome (GEO) metric, a new method for identifying potential partisan gerrymanders. In contrast with currently popular methods, the GEO metric uses both geographic information about a districting plan as well as district-level partisan data, rather than just one or the other. We motivate and define the GEO metric, which gives a count (a non-negative integer) to each political party. The count indicates the number of previously lost districts which that party potentially could have had a 50% chance of winning, without risking any currently won districts, by making reasonable changes to the input map. We then analyze GEO metric scores for each party in several recent elections. We show that this relatively easy to understand and compute metric can encapsulate the results from more elaborate analyses., Comment: 22 pages, 18 figures/tables. Maps analyses and related code available at https://www.the-geometric.com/ To be published in Election Law Journal

The vertex distortion of a lattice knot is the supremum of the ratio of the distance between a pair of vertices along the knot and their distance in the l1-norm. We show analogous results to those of Gromov, Pardon and Blair-Campisi-Taylor-Tomova about the distortion of smooth knots hold for vertex distortion: the vertex distortion of a lattice knot is 1 only if it is the unknot, and that there are minimal lattice-stick number knot conformations with arbitrarily high distortion.

We adapt work of Kirby-Thompson and Zupan to define an integer invariant $\mathcal{L}(\mathcal{T})$ of a bridge trisection $\mathcal{T}$ of a smooth surface $\mathcal{K}$ in $S^4$ or $B^4$. We show that when $\mathcal{L}(\mathcal{T})=0$, then the surface $\mathcal{K}$ is unknotted. We also show show that for a trisection $\mathcal{T}$ of an irreducible surface, bridge number produces a lower bound for $\mathcal{L}(\mathcal{T})$. Consequently, $\mathcal{L}$ can be arbitrarily large.

We show that the disk complex of a genus $g>1$ Heegaard surface for the 3-sphere is homotopy equivalent to a wedge of $(2g-2)$-dimensional spheres. This implies that genus $g>1$ Heegaard surfaces for the 3-sphere are topologically minimal with index $2g-1$.

We explore the Declination, a new metric intended to detect partisan gerrymandering. We consider instances in which each district has equal turnout, the maximum turnout to minimum turnout is bounded, and turnout is unrestricted. For each of these cases, we show exactly which vote-share, seat-share pairs $(V,S)$ have an election outcome with Declination equal to 0. We also show how our analyses can be applied to finding vote-share, seat-share pairs that are possible for nonzero Declination. Within our analyses, we show that Declination cannot detect all forms of packing and cracking, and we compare the Declination to the Efficiency Gap. We show that these two metrics can behave quite differently, and give explicit examples of that occurring.

If a graph is in bridge position in a 3-manifold so that the graph complement is irreducible and boundary irreducible, we generalize a result of Bachman and Schleimer to prove that the complexity of a surface properly embedded in the complement of the graph bounds the graph distance of the bridge surface. We use this result to construct, for any natural number $n$, a hyperbolic manifold containing a surface of topological index $n$., Comment: 12 pages

We extend techniques due to Pardon to show that there is a lower bound on the distortion of a knot in $\mathbb{R}^3$ proportional to the minimum of the bridge distance and the bridge number of the knot. We also exhibit an infinite family of knots for which the minimum of the bridge distance and the bridge number is unbounded and Pardon's lower bound is constant., Comment: 19 pages, 5 figures. Version 2 incorporates significant improvements in exposition. The paper has been accepted by Journal of Topology

We show that every knot is one crossing change away from a knot of arbitrarily high bridge number and arbitrarily high bridge distance., Comment: Accepted by American Mathematical Monthly. New version incorporates referee comments

In this paper, we characterize all links in the 3-sphere with bridge number at least three that have a bridge sphere of distance two. We show that a link L has a bridge sphere of distance at most two then it falls into at least one of three categories: (1) The exterior of L contains an essential meridional sphere. (2) L can be decomposed as a tangle product of a Montesinos tangle with an essential tangle in a way that respects the bridge surface and either the Montesinos tangle is rational or the essential tangle contains an incompressible, boundary-incompressible annulus. (3) L is obtained by banding from another link L' that has a bridge sphere of the same Euler characteristic as the bridge sphere for L but of distance 0 or 1., Comment: 27 pages, 13 figures

If a knot K in a closed, orientable 3-manifold M has a bridge surface T with distance at least 3 in the curve complex of T - K, then the genus of any essential surface in its exterior with non-empty, non-meridional boundary gives rise to an upper bound for the bridge number of K with respect to T. In particular, a nontrivial, aspherical, and atoroidal knot K with such a bridge surface has its bridge number bounded by 5 if K has a non-trivial reducing surgery; 6 if K has a non-trivial toroidal surgery; and 4g + 2 if K is null-homologous and has Seifert genus g., Comment: 9 pages

We show that the distance of a link $K$ with respect to a bridge surface of any genus determines a lower bound on the genus of essential surfaces and Heegaard surfaces in the manifolds that result from non-trivial Dehn surgeries on the knot. In particular, knots with high bridge distance do not admit non-trivial non-hyperbolic surgeries or non-trivial cosmetic surgeries. We further show that if a knot has bridge distance at least 3 then its bridge number is bounded above by a function of Seifert genus, or indeed by the genus of (almost) any essential surface or Heegaard surface in the surgered manifold., Comment: The new version strengthens the results, incorporates arXiv:1211.4787 and incorporates referee comments. 50 pages, 8 figures

We generalize the definition of thin position of Scharlemann and Thompson for compact orientable 3-manifolds with torus boundary components and introduce $\alpha$-sloped generalized Heegaard splittings. We examine its relationship to generalized Heegaard splittings of manifolds resulting from Dehn filling. We compare alpha-sloped thin position of 3-manifolds to other types of thin position for knots and 3-manifolds and discuss how this kind of decomposition gives an organic picture of M and allows the structure of the manifold to dictate the most natural slope(s) on the boundary. Additionally, we provide illustrative examples and questions motivating the study of alpha-sloped thin position., Comment: 21 pages, 7 figures; Revisions to theorem 3.4. The revisions do not affect theorem 1.1

Let M be a closed 3-manifold with a given Heegaard splitting. We show that after a single stabilization, some core of the stabilized splitting has arbitrarily high distance with respect to the splitting surface. This generalizes a result of Minsky, Moriah, and Schleimer for knots in S^3. We also show that in the complex of curves, handlebody sets are either coarsely distinct or identical. We define the coarse mapping class group of a Heeegaard splitting, and show that if (S, V, W) is a Heegaard splitting of genus greater than or equal to 2, then the coarse mapping class group of (S,V,W) is isomorphic to the mapping class group of (S, V,W)., Comment: Certain misstatements about the pair of pants decompositions constructed for reducible Heegaard splittings have been corrected. The paper has also been restructured some to aid the exposition of the proof. Details have been provided for the end of the proof of the main theorem. And the first author's name has been changed