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Scott A. Taylor

Publications and source records attributed to Scott A. Taylor.

At least 19 recordsLinked to original sources
arXiv2025

Hyperbolic Brunnian Theta Curves

A nontrivial $θ$-curve in $S^3$ is Brunnian if each of its cycles is the unknot. We show that if the exterior of a Brunnian $θ$-curve is atoroidal, then it does not contain an essential annulus. Previously, Ozawa-Tsutsumi showed that there is no essential disc. Consequently, by Thurston's work, the exterior of an atoroidal Brunnian $θ$-curve is hyperbolic with totally geodesic boundary. It follows that Brunnian $θ$-curves of low bridge number have exteriors that are hyperbolic with totally geodesic boundary. We also show that two Brunnian $θ$-curves are isotopic if and only if they are neighborhood isotopic and classify Brunnian spines of genus 2 handlebody knots. We rely heavily on a classification of annuli in the exteriors of genus two handlebody knots by Koda-Ozawa and further developed by Wang in conjunction with sutured manifold theory results of Taylor.

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arXiv2025

The genus 1 bridge number of satellite knots

Let $T$ be a satellite knot, link, or spatial graph in a 3-manifold $M$ that is either $S^3$ or a lens space. Let $\mathfrak{b}_0$ and $\mathfrak{b}_1$ denote genus 0 and genus 1 bridge number, respectively. Suppose that $T$ has a companion knot $K$ (necessarily not the unknot) and wrapping number $ω$ with respect to $K$. When $K$ is not a torus knot, we show that $\mathfrak{b}_1(T)\geq ω\mathfrak{b}_1(K)$. There are previously known counter-examples if $K$ is a torus knot. Along the way, we generalize and give a new proof of Schubert's result that $\mathfrak{b}_0(T) \geq ω\mathfrak{b}_0(K)$. We also prove versions of the theorem applicable to when $T$ is a ``lensed satellite'' and when there is a torus separating components of $T$.

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arXiv2025

Closures of 1-tangles and annulus twists

A 1-tangle is a properly embedded arc $ψ$ in an unknotted solid torus $V$ in $S^3$. Attaching an arc $ϕ$ in the complementary solid torus $W$ to its endpoints creates a knot $K(ϕ)$ called the closure of $ψ$. We show that for a given nontrivial 1-tangle $ψ$ there exist at most two closures that are the unknot. We give a general method for producing nontrivial 1-tangles admitting two distinct closures and show that our construction accounts for all such examples. As an application, we show that if we twist an unknot $q \neq 0$ times around an unknotted sufficiently incompressible annulus intersecting it exactly once, then there is at most one $q$ such that the resulting knot is unknotted and, if there is such, then $q = \pm 1$. With additional work, we also show that the Krebes 1-tangle does not admit an unknot closure. Our key tools are the ``wrapping index'' which compares how two complementary 1-tangles $ϕ_1$ and $ϕ_2$ wrap around $W$, a theorem of the author's from sutured manifold theory, and theorems of Gabai and Scharlemann concerning band sums.

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arXiv2022

Equivariant Heegaard genus of reducible 3-manifolds

The equivariant Heegaard genus of a 3-manifold $M$ with the action of a finite group $G$ of diffeomorphisms is the smallest genus of an equivariant Heegaard splitting for $M$. Although a Heegaard splitting of a reducible manifold is reducible and although if $M$ is reducible, there is an equivariant essential sphere, we show that equivariant Heegaard genus may be super-additive, additive, or sub-additive under equivariant connected sum. Using a thin position theory for 3-dimensional orbifolds, we establish sharp bounds on the equivariant Heegaard genus of reducible manifolds, similar to those known for tunnel number.

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arXiv2021

Bounding the Kirby-Thompson invariant of spun knots

A bridge trisection of a smooth surface in $S^4$ is a decomposition analogous to a bridge splitting of a link in $S^3$. The Kirby-Thompson invariant of a bridge trisection measures its complexity in terms of distances between disc sets in the pants complex of the trisection surface. We give the first significant bounds for the Kirby-Thompson invariant of spun knots. In particular, we show that the Kirby-Thompson invariant of the spun trefoil is 15.

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arXiv2021

Links, bridge number, and width trees

To each link $L$ in $S^3$ we associate a collection of certain labelled directed trees, called width trees. We interpret some classical and new topological link invariants in terms of these width trees and show how the geometric structure of the width trees can bound the values of these invariants from below. We also show that each width tree is associated with a knot in $S^3$ and that if it also meets a high enough "distance threshold" it is, up to a certain equivalence, the unique width tree realizing the invariants.

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arXiv2021

Kirby-Thompson distance for trisections of knotted surfaces

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.

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arXiv2021

Tunnel number and bridge number of composite genus 2 spatial graphs

Connected sum and trivalent vertex sum are natural operations on genus 2 spatial graphs and, as with knots, tunnel number behaves in interesting ways under these operations. We prove sharp Scharlemann-Schultens type bounds for the tunnel number of a composite genus 2 spatial graph. For the tunnel number of a composite Brunnian $θ$-curve, our result implies that the tunnel number is at least the number of summands, as in the knot case. We also prove a version of a theorem of Morimoto for knots: the tunnel number of a composite m-small genus 2 spatial graph is at least the sum of the tunnel numbers of the factors. We also study lower bounds for the bridge number of composite genus 2 graphs. In particular, our results imply that for a Brunnian composite $θ$-curve having $m$ factors in its prime factorization, the bridge number is at least $m+3/2$.

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arXiv2019

Distortion and the bridge distance of knots

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.

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arXiv2019

Combinatorial minimal surfaces in pseudomanifolds

We define combinatorial analogues of stable and unstable minimal surfaces in the setting of weighted pseudomanifolds. We prove that, under mild conditions, such combinatorial minimal surfaces always exist. We use a technique, adapted from work of Johnson and Thompson, called thin position. Thin position is defined using orderings of the cells of a pseudomanifold. In addition to defining and finding combinatorial minimal surfaces, from thin orderings, we derive invariants of even-dimensional closed simplicial pseudomanifolds called width and trunk. We study additivity properties of these invariants under connected sum and prove theorems analogous to those in knot theory and 3-manifold theory.

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arXiv2019

Abstractly Planar Spatial Graphs

This is a survey article for the forthcoming `A Concise Encyclopedia of Knot Theory.' We focus on the topology of spatial graphs with few vertices and edges, paying particular attention to Brunnian $θ$-graphs.

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arXiv2018

Two more proofs that the Kinoshita graph is knotted

The Kinoshita graph is a particular embedding in the 3-sphere of a graph with three edges, two vertices and no loops. It has the remarkable property that although the removal of any edge results in an unknotted loop, the Kinoshita graph is itself knotted. We use two classical theorems from knot theory to give two particularly simple proofs that the Kinoshita graph is knotted. Apart from appealing to the two classical theorems, the exposition is elementary and is aimed at those with only a passing familiarity with knot theory.

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arXiv2018

Thin position for knots, links, and graphs in 3-manifolds

We define a new notion of thin position for a graph in a 3-manifold which combines the ideas of thin position for manifolds first originated by Scharlemann and Thompson with the idea of thin position for knots first originated by Gabai. This thin position has the property that connect summing annuli and pairs-of-pants show up as thin levels. In a forthcoming paper, this new thin position allows us to define two new families of invariants of knots, links, and graphs in 3-manifolds. The invariants in one family are similar to bridge number and the invariants in the other family are similar to Gabai's width for knots in the 3-sphere. The invariants in both families detect the unknot and are additive under connected sum and trivalent vertex sum.

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arXiv2018

Additive invariants for knots, links and graphs in 3-manifolds

We define two new families of invariants for (3-manifold, graph) pairs which detect the unknot and are additive under connected sum of pairs and (-1/2)-additive under trivalent vertex sum of pairs. The first of these families is closely related to both bridge number and tunnel number. The second of these families is a variation and generalization of Gabai's width for knots in the 3-sphere. We give applications to the tunnel number and higher genus bridge number of connected sums of knots.

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arXiv2016

Dehn filling and the Thurston norm

For a compact, orientable, irreducible 3-manifold with toroidal boundary that is not the product of a torus and an interval or a cable space, each boundary torus has a finite set of slopes such that, if avoided, the Thurston norm of a Dehn filling behaves predictably. More precisely, for all but finitely many slopes, the Thurston norm of a class in the second homology of the filled manifold plus the so-called winding norm of the class will be equal to the Thurston norm of the corresponding class in the second homology of the unfilled manifold. This generalizes a result of Sela and is used to answer a question of Baker-Motegi concerning the Seifert genus of knots obtained by twisting a given initial knot along an unknot which links it.

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arXiv2016

Exceptional and cosmetic surgeries on knots

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.

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arXiv2015

New examples of Brunnian theta graphs

The Kinoshita graph is the most famous example of a Brunnian theta graph, a nontrivial spatial theta graph with the property that removing any edge yields an unknot. We produce a new family of diagrams of spatial theta graphs with the property that removing any edge results in the unknot. The family is parameterized by a certain subgroup of the pure braid group on four strands. We prove that infinitely many of these diagrams give rise to non-isotopic Brunnian theta graphs.

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