arXiv subjects

Jessica S. Purcell

Publications and source records attributed to Jessica S. Purcell.

At least 19 recordsLinked to original sources

arXiv2026

Hyperbolic manifolds without positive spun triangulations

Using a result of Choi, we provide the first examples of pairs consisting of a closed hyperbolic three-manifold and a simple closed geodesic, such that there is no positive spun ideal triangulation for the manifold, spun about the chosen geodesic. In our first two examples, the closed manifold is the third manifold in the SnapPy census, also known as Vol3, and the geodesics are its systole and second systole. This provides evidence for the conjecture that Vol3 has no positive spun ideal triangulation for any choice of geodesic.

math.GT↗

arXiv2026

Volumes of ideal hyperbolic drums

Milnor computed the volumes of ideal hyperbolic prisms as part of an effort to construct 3-manifolds whose volumes are finite rational sums of the Lobachevsky function evaluated at rational multiples of pi. Motivated by these results and with an eye to related applications, we prove a volume formula for arbitrary ideal hyperbolic antiprisms, also called drums.

math.GT↗

arXiv2025

Excluding cosmetic surgeries on hyperbolic 3-manifolds

This paper employs knot invariants and results from hyperbolic geometry to develop a practical procedure for checking the cosmetic surgery conjecture on any given one-cusped manifold. This procedure has been used to establish the following computational results. First, we verify that all knots up to 19 crossings, and all one-cusped 3-manifolds in the SnapPy census, do not admit any purely cosmetic surgeries. Second, we check that a hyperbolic knot with at most 15 crossings only admits chirally cosmetic surgeries when the knot itself is amphicheiral. Third, we enumerate all knots up to 13 crossings that share a common Dehn fillings with the figure-8 knot. The code that verifies these results is publicly available on GitHub.

math.GT↗

arXiv2025

Polynomial bounds for surfaces in cusped 3-manifolds

It is natural to ask how many isotopy classes of embedded essential surfaces lie in a given 3-manifold. The first bounds on the number of such surfaces were exponential, using normal surfaces. More recently, by restricting to alternating link complements in 3-sphere, Hass, Thompson and Tsvietkova obtained polynomial bounds, but for a limited class of surfaces: closed and spanning ones. Here, we complete the picture for classical alternating links and extend these results to other classes of cusped 3-manifolds. We give explicit polynomial bounds on all embedded essential surfaces, closed or any boundary slope, orientable or non-orientable. Our 3-manifolds are complements of links with alternating diagrams on wide classes of surfaces in broad families of 3-manifolds. This includes all alternating links in 3-sphere as well as many non-alternating ones, alternating virtual knots, many toroidally alternating knots, and most Dehn fillings of such manifolds.

math.GT↗

arXiv2025

Rigidity of highly twisted plat diagrams

In this paper we prove that if a knot or link has a sufficiently complicated plat projection, then that plat projection is unique. More precisely, if a knot or link has a $2m$-plat projection, where $m$ is at least four, and height at least two, and each twist region of the plat contains at least four crossings, then such a projection is unique up to obvious rotations. In particular, this projection gives a canonical form for such knots and links, and thus provides a classification of these links.

math.GT↗

arXiv2025

Volume bounds for hyperbolic rod complements in the 3-torus

The study of rod complements is motivated by rod packing structures in crystallography. We view them as complements of links comprised of Euclidean geodesics in the 3-torus. Recent work of the second author classifies when such rod complements admit hyperbolic structures, but their geometric properties are yet to be well understood. In this paper, we provide upper and lower bounds for the volumes of all hyperbolic rod complements in terms of rod parameters, and show that these bounds may be loose in general. We introduce better and asymptotically sharp volume bounds for a family of rod complements. The bounds depend only on the lengths of the continued fractions formed from the rod parameters.

math.GT↗

arXiv2025

Augmented links, shadow links, and the TV volume conjecture: a geometric perspective

For hyperbolic 3-manifolds, the growth rate of their Turaev-Viro invariants, evaluated at a certain root of unity, is conjectured to give the hyperbolic volume of the manifold. This has been verified for a handful of examples and several infinite families of link complements, including fundamental shadow links. Fundamental shadow links lie in connected sums of copies of $S^1\times S^2$, and their complements are built of regular ideal octahedra. Another well-known family of links with complements built of regular ideal octahedra are the octahedral fully augmented links in the 3-sphere. The complements of these links are now known to be homeomorphic to complements of fundamental shadow links, using topological techniques. In this paper, we give a new, geometric proof that complements of octahedral fully augmented links are isometric to complements of fundamental shadow links. We then use skein theoretic techniques to determine formulae for coloured Jones polynomials of these links. In the case of no half-twists, this gives a new, more geometric verification of the Turaev-Viro volume conjecture for these links.

math.GT↗

arXiv2025

On Geometric triangulations of double twist knots

In this paper we construct two different explicit triangulations of the family of double twist knots $K(p,q)$ using methods of triangulating Dehn fillings, with layered solid tori and their double covers. One construction yields the canonical triangulation, and one yields a triangulation that we conjecture is minimal. We prove that both are geometric, meaning they are built of positively oriented convex hyperbolic tetrahedra. We use the conjecturally minimal triangulation to present eight equations cutting out the A-polynomial of these knots.

math.GT↗

arXiv2025

Geometry of fully augmented links in doubled 3-manifolds

Classical fully augmented links have explicit hyperbolic geometry, and have diagrams on the 2-sphere in the 3-sphere. We generalise to construct fully augmented links projected to the reflection surface of any 3-manifold obtained by doubling a compact 3-manifold and show that the results of the classical setting extend to these links. When the resulting manifolds are hyperbolic, we find bounds on their cusp shapes and volumes. Note these links include virtual fully augmented links, and thus our bounds apply to such links when they are hyperbolic.

math.GT↗

arXiv2024

Alternating links on nonorientable surfaces and Klein-bottly alternating links

It has been known for several decades that classical alternating links in the 3-sphere have nice hyperbolic geometric properties. Recent work generalises such results to give hyperbolic geometry of links with alternating projections onto any surface in very general 3-manifolds. However, the most general results require an orientable projection surface. In this paper, we extend to alternating links on nonorientable projection surfaces. As an application, we study Klein bottly alternating links in prism manifolds, which are a natural generalisation of Adams' toroidally alternating links in lens spaces.

math.GT↗

arXiv2024

Triangulations of the 3-sphere with knotted edge

We prove that for any knot $K$, there exists a one-vertex triangulation of the $3$-sphere containing an edge forming $K$. The proof is constructive, and based on fully augmented links. We use our method to produce ``complicated'' simplicial triangulations of the $3$-sphere that we show are smallest possible, up to a constant multiplicative factor.

math.GT↗

arXiv2024

Hyperbolic and satellite Lorenz links obtained by twisting

A Lorenz link is equivalent to a T-link, which is a positive braid built by concatenating torus braids of increasing size. When each torus braid except the largest is obtained by full twists, then the T-link can be described as the Dehn filling of a parent link. In this paper, we completely classify when such parent links are hyperbolic. This gives a classification of the geometry of T-links obtained by full twists when the amount of twisting is large, although the bound on the number of required twists is not effective. We also present effective results on hyperbolicity for two families of T-links obtained by twisting, even when the number of twists is small. Finally, we identify families of satellite T-links obtained by half-twists.

math.GT↗

arXiv2024

A-polynomials, Ptolemy equations and Dehn filling

The A-polynomial encodes hyperbolic geometric information on knots and related manifolds. Historically, it has been difficult to compute, and particularly difficult to determine A-polynomials of infinite families of knots. Here, we compute A-polynomials by starting with a triangulation of a manifold, then using symplectic properties of the Neumann-Zagier matrix encoding the gluings to change the basis of the computation. The result is a simplification of the defining equations. We apply this method to families of manifolds obtained by Dehn filling, and show that the defining equations of their A-polynomials are Ptolemy equations which, up to signs, are equations between cluster variables in the cluster algebra of the cusp torus.

math.GT↗

arXiv2024

Arithmetic modular links

We construct a sequence of geodesics on the modular surface such that the complement of the canonical lifts to the unit tangent bundle are arithmetic 3-manifolds.

math.GT↗

arXiv2023

Standard position for surfaces in link complements in arbitrary 3-manifolds

Since the 1980s, it has been known that essential surfaces in alternating link complements can be isotoped to be transverse to the link diagram almost everywhere, with the exception of some well-understood intersections, and described combinatorially as a result. This was called standard position for surfaces and has had numerous applications. However, the original techniques only apply to classical alternating links projected onto the 2-sphere inside the 3-sphere. In this paper, we prove that standard position for surfaces can be extended to a broader class, namely weakly generalized alternating links. Such links include all classical prime non-split alternating links in the 3-sphere, and also many links that are alternating on higher genus surfaces, or lie in manifolds besides the 3-sphere. As an application, we show that all such links are prime, and that under mild restrictions, essential Conway spheres for such links interact with the diagram exactly as in the classical alternating setting.

math.GT↗

arXiv2023

A-Polynomials of fillings of the Whitehead sister

Knots obtained by Dehn filling the Whitehead sister link include some of the smallest volume twisted torus knots. Here, using results on A-polynomials of Dehn fillings, we give formulas to compute the A-polynomials of these knots. Our methods also apply to more general Dehn fillings of the Whitehead sister.

math.GT↗

arXiv2023

On the geometry of rod packings in the 3-torus

Rod packings in the 3-torus encode information of some crystal structures in crystallography. They can be viewed as links in the 3-torus, and tools from 3-manifold geometry and topology can be used to study their complements. In this paper, we initiate the use of geometrisation to study such packings. We analyse the geometric structures of the complements of simple rod packings, and find families that are hyperbolic and Seifert fibred.

math.GT↗

arXiv2023

Large volume fibred knots of fixed genus

We show that, for hyperbolic fibred knots in the three-sphere, the volume and the genus are unrelated. Furthermore, for such knots, the volume is unrelated to strong quasipositivity and Seifert form.

math.GT↗