Scott A. Taylor

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math.GT2025

Hyperbolic Brunnian Theta Curves

Luis Celso Chan Palomo, Scott A. Taylor

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 esse…

math.GT2025

Closures of 1-tangles and annulus twists

Scott A. Taylor

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 $K (ϕ)$ …

math.GT2024

Signature, slicing foams, and crossing changes of Klein graphs

Kenneth L. Baker, Allison H. Moore, Danielle O'Donnol +1

A totally oriented Klein graph is a trivalent spatial graph in the 3-sphere with a 3-coloring of its edges and an orientation on each bicolored link. A totally oriented Klein foam…

math.GT2024

Strong Haken via Thin Position

Scott Taylor

We use thin position of Heegaard splittings to give a new proof of Haken's Lemma that a Heegaard surface of a reducible manifold is reducible and of Scharlemann's ``Strong Haken Th…