Birch Bryant | Searcharxiv

arXiv2026

Multivariable Alexander polynomials of twisted torus links via Dehn surgery

Let $T(p,q;r,s)$ be the twisted torus link obtained from the torus link $T(p,q)$ by adding $s$ full twists to $r$ adjacent strands, and let $C$ be an unknotted circle encircling those strands. For all $p$, $q$, and $r$ we compute the multivariable Alexander polynomial of $T(p,q)\cup C$ in closed form. A relation of Silver and Williams between the Alexander polynomials of a link and of its $1/n$ Dehn fillings then gives a closed formula for the multivariable Alexander polynomial of every twisted torus link $T(p,q;r,s)$, answering a question of Adnan and Park. When $\gcd(p,q)=1$, so that $T(p,q;r,s)$ is a knot, the formula is a sum of $pr$ monomials with coefficient $+1$, divided by cyclotomic factors, and its exponents are governed by an elementary counting function on the orbit of $q$ modulo $p$. Because the numerator has no cancellation, we are able to give the degree for all $s$ and a simple non-fiberedness criterion.

math.GT↗

arXiv2025

Ideal Triangulations and Once-Punctured Surface Bundles

A well-known result of Walsh states that if $\mathcal T^*$ is an ideal triangulation of an atoroidal, acylindrical, irreducible, compact 3-manifold with torus boundary components, then every properly embedded, two-sided, incompressible surface $S$ is isotopic to a spun-normal surface unless $S$ is isotopic to a fiber or virtual fiber. Previously it was unknown if for such a 3-manifold an ideal triangulation in which a fiber spun-normalizes exists. We give a proof of existence and give an algorithm to construct the ideal triangulation provided the 3-manifold has a single boundary component.

math.GT↗

arXiv2020

Efficient triangulations and boundary slopes

For a compact, irreducible, $\partial$-irreducible, an-annular bounded 3-manifold $M\ne\mathbb{B}^3$, then any triangulation $\mathcal{T}$ of $M$ can be modified to an ideal triangulation $\mathcal{T}^*$ of $\stackrel{\circ}{M}$. We use the inverse relationship of crushing a triangulation along a normal surface and that of inflating an ideal triangulation to introduce and study boundary-efficient triangulations and end-efficient ideal triangulations. We prove that the topological conditions necessary for a compact 3-manifold $M$ admitting an annular-efficient triangulation are sufficient to modify any triangulation of $M$ to a boundary-efficient triangulation which is also annular-efficient. From the proof we have for any ideal triangulation $T^*$ and any inflation $\mathcal{T}_Λ$, there is a bijective correspondence between the closed normal surfaces in $\mathcal{T}^*$ and the closed normal surfaces in $\mathcal{T}_Λ$ with corresponding normal surfaces being homeomorphic. It follows that for an ideal triangulation $\mathcal{T}^*$ that is $0$-efficient, $1$-efficient, or end-efficient, then any inflation $\mathcal{T}_Λ$ of $\mathcal{T}^*$ is $0$-efficient, $1$-efficient, or $\partial$-efficient, respectively. There are algorithms to decide if a given triangulation or ideal triangulation of a $3$-manifold is one of these efficient triangulations. Finally, it is shown that for an annular-efficient triangulation, there are only a finite number of boundary slopes for normal surfaces of a bounded Euler characteristic; hence, in a compact, orientable, irreducible, $\partial$-irreducible, and an-annular $3$-manifold, there are only finitely many boundary slopes for incompressible and $\partial$-incompressible surfaces of a bounded Euler characteristic.

math.GT↗