Hyperbolic Knots Research Articles

An infinite family of chiral, non-slice, non-alternating hyperbolic knots with vanishing Upsilon invariants

For a knot in the 3-sphere, its Upsilon invariant is a continuous, piecewise linear function defined on the interval [0,2]. It is a concordance invariant, so vanishes for slice knots. Since it only changes the sign for the mirror image with reversed orientation, it also vanishes for amphicheiral knots. For alternating knots, more generally, quasi-alternating knots, the Upsilon invariant is determined only by the signature. In particular, it vanishes if such a knot has zero signature. On the other hand, it is known that the Upsilon invariant is a non-zero convex function for L-space knots. In this paper, we construct an infinite family of hyperbolic knots, each of which has zero Upsilon invariant, but is chiral, non-slice, non-alternating. To confirm that the Upsilon invariant vanishes, we calculate the full knot Floer complex.

The signature and cusp geometry of hyperbolic knots

The signature and cusp geometry of hyperbolic knots

Knots, Perturbative Series and Quantum Modularity

We introduce an invariant of a hyperbolic knot which is a map $\alpha\mapsto \boldsymbol{\Phi}_\alpha(h)$ from $\mathbb{Q}/\mathbb{Z}$ to matrices with entries in $\overline{\mathbb{Q}}[[h]]$ and with rows and columns indexed by the boundary parabolic ${\rm SL}_2(\mathbb{C})$ representations of the fundamental group of the knot. These matrix invariants have a rich structure: (a) their $(\sigma_0,\sigma_1)$ entry, where $\sigma_0$ is the trivial and $\sigma_1$ the geometric representation, is the power series expansion of the Kashaev invariant of the knot around the root of unity ${\rm e}^{2\pi{\rm i} \alpha}$ as an element of the Habiro ring, and the remaining entries belong to generalized Habiro rings of number fields; (b) the first column is given by the perturbative power series of Dimofte-Garoufalidis; (c) the columns of $\boldsymbol{\Phi}$ are fundamental solutions of a linear $q$-difference equation; (d) the matrix defines an ${\rm SL}_2(\mathbb{Z})$-cocycle $W_\gamma$ in matrix-valued functions on $\mathbb{Q}$ that conjecturally extends to a smooth function on $\mathbb{R}$ and even to holomorphic functions on suitable complex cut planes, lifting the factorially divergent series $\boldsymbol{\Phi}(h)$ to actual functions. The two invariants $\boldsymbol{\Phi}$ and $W_\gamma$ are related by a refined quantum modularity conjecture which we illustrate in detail for the three simplest hyperbolic knots, the $4_1$, $5_2$ and $(-2,3,7)$ pretzel knots. This paper has two sequels, one giving a different realization of our invariant as a matrix of convergent $q$-series with integer coefficients and the other studying its Habiro-like arithmetic properties in more depth.

Harer–Zagier transform of the HOMFLY–PT polynomial for families of twisted hyperbolic knots

In an attempt to generalise knot matrix models for non-torus knots, which currently remains an open problem, we derived expressions for the Harer–Zagier transform—a discrete Laplace transform—of the HOMFLY–PT polynomial for some infinite families of twisted hyperbolic knots. Among them, we found a family of pretzel knots for which, like for torus knots, the transform has a fully factorised form, while for the remaining families considered it consists of sums of factorised terms. Their zero loci show a remarkable structure, which mostly lies on the unit circle and deviates from it only in pairs.

On the number of nested twice-punctured tori in a hyperbolic knot exterior

This paper continues a program due to Motegi regarding universal bounds for the number of nonisotopic essential n-punctured tori in the complement of a hyperbolic knot in S3. For n=1, Valdez-Sánchez showed that there are at most five nonisotopic Seifert tori in the exterior of a hyperbolic knot. In this paper, we address the case n=2. We show that there are at most six nonisotopic, nested, essential 2-holed tori in the complement of every hyperbolic knot.

Dehn Fillings of Knot Manifolds Containing Essential Twice-Punctured Tori

We show that if a hyperbolic knot manifold M M contains an essential twice-punctured torus F F with boundary slope β \beta and admits a filling with slope α \alpha producing a Seifert fibred space, then the distance between the slopes α \alpha and β \beta is less than or equal to 5 5 unless M M is the exterior of the figure eight knot. The result is sharp; the bound of 5 5 can be realized on infinitely many hyperbolic knot manifolds. We also determine distance bounds in the case that the fundamental group of the α \alpha -filling contains no non-abelian free group. The proofs are divided into the four cases F F is a semi-fibre, F F is a fibre, F F is non-separating but not a fibre, and F F is separating but not a semi-fibre, and we obtain refined bounds in each case.

Euclidean volumes of hyperbolic knots

The hyperbolic structure on a 3 3 –dimensional cone–manifold with a knot as singularity can often be deformed into a limiting Euclidean structure. In the present paper we show that the respective normalised Euclidean volume is always an algebraic number, which is reminiscent of Sabitov’s theorem (the Bellows Conjecture). This fact also stands in contrast to hyperbolic volumes whose number–theoretic nature is usually quite complicated.

Knots and Their Related $q$-Series

We discuss a matrix of periodic holomorphic functions in the upper and lower half-plane which can be obtained from a factorization of an Andersen-Kashaev state integral of a knot complement with remarkable analytic and asymptotic properties that defines a ${\rm PSL}_2({\mathbb Z})$-cocycle on the space of matrix-valued piecewise analytic functions on the real numbers. We identify the corresponding cocycle with the one coming from the Kashaev invariant of a knot (and its matrix-valued extension) via the refined quantum modularity conjecture of [arXiv:2111.06645] and also relate the matrix-valued invariant with the 3D-index of Dimofte-Gaiotto-Gukov. The cocycle also has an analytic extendability property that leads to the notion of a matrix-valued holomorphic quantum modular form. This is a tale of several independent discoveries, both empirical and theoretical, all illustrated by the three simplest hyperbolic knots.

Twisted Neumann–Zagier matrices

The Neumann–Zagier matrices of an ideal triangulation are integer matrices with symplectic properties whose entries encode the number of tetrahedra that wind around each edge of the triangulation. They can be used as input data for the construction of a number of quantum invariants that include the loop invariants, the 3D-index and state-integrals. We define a twisted version of Neumann–Zagier matrices, describe their symplectic properties, and show how to compute them from the combinatorics of an ideal triangulation. As a sample application, we use them to define a twisted version of the 1-loop invariant (a topological invariant) which determines the 1-loop invariant of the cyclic covers of a hyperbolic knot complement, and conjecturally is equal to the adjoint twisted Alexander polynomial.

Constructing knots with specified geometric limits

It is known that any tame hyperbolic 3-manifold with infinite volume and a single end is the geometric limit of a sequence of finite volume hyperbolic knot complements. Purcell and Souto showed that if the original manifold embeds in the 3-sphere, then such knots can be taken to lie in the 3-sphere. However, their proof was nonconstructive; no examples were produced. In this paper, we give a constructive proof in the geometrically finite case. That is, given a geometrically finite, tame hyperbolic 3-manifold with one end, we build an explicit family of knots whose complements converge to it geometrically. Our knots lie in the (topological) double of the original manifold. The construction generalises the class of fully augmented links to a Kleinian groups setting.

Spines and surgery descriptions of graph manifolds

Graph manifolds are compact orientable 3–manifolds obtained by gluing several copies of D2×S1 and N2×S1 together by homeomorphisms of some components of their boundaries (D2 is the 2–disc and N2 denotes the 2–disc with two holes). Here we study spines and surgery representations of orientable graph manifolds, and derive geometric presentations of their fundamental group. Then we determine the homeomorphism type of many Takahashi manifolds and the Teragaito manifolds, showing that they are graph manifolds with specified invariants. Finally, we describe graph manifolds arising from toroidal surgeries on certain classes of hyperbolic knots.

Topological Quantum Computation is Hyperbolic

We show that a topological quantum computer based on the evaluation of a Witten–Reshetikhin–Turaev TQFT invariant of knots can always be arranged so that the knot diagrams with which one computes are diagrams of hyperbolic knots. The diagrams can even be arranged to have additional nice properties, such as being alternating with minimal crossing number. Moreover, the reduction is polynomially uniform in the self-braiding exponent of the coloring object. Various complexity-theoretic hardness results regarding the calculation of quantum invariants of knots follow as corollaries. In particular, we argue that the hyperbolic geometry of knots is unlikely to be useful for topological quantum computation.

An Enhanced Euler Characteristic of Sutured Instanton Homology

Abstract For a balanced sutured manifold $(M,\gamma )$, we construct a decomposition of $SHI(M,\gamma )$ with respect to torsions in $H=H_{1}(M;\mathbb {Z})$, which generalizes the decomposition of $I^{\sharp }(Y)$ in a previous work of the authors. This decomposition can be regarded as a candidate for the counterpart of the torsion spin$^{c}$ decompositions in $SFH(M,\gamma )$. Based on this decomposition, we define an enhanced Euler characteristic $\chi _{\textrm {en}}(SHI(M,\gamma ))\in \mathbb {Z}[H]/\pm H$ and prove that $\chi _{\textrm {en}}(SHI(M,\gamma ))=\chi (SFH(M,\gamma ))$. This provides a better lower bound on $\dim _{\mathbb {C}}SHI(M,\gamma )$ than the graded Euler characteristic $\chi _{\textrm {gr}}(SHI(M,\gamma ))$. As applications, we prove instanton knot homology detects the unknot in any instanton L-space and show that the conjecture $KHI(Y,K)\cong \widehat {HFK}(Y,K)$ holds for all $(1,1)$-L-space knots and constrained knots in lens spaces, which include all torus knots and many hyperbolic knots in lens spaces.

Distinguishing some genus one knots using finite quotients

We give a criterion for distinguishing a prime knot [Formula: see text] in [Formula: see text] from every other knot in [Formula: see text] using the finite quotients of [Formula: see text]. Using recent work of Baldwin–Sivek, we apply this criterion to the hyperbolic knots [Formula: see text], [Formula: see text], and the three-strand pretzel knots [Formula: see text] for every integer [Formula: see text].

Torus Lorenz links obtained by full twists along torus links

All knots are known to be hyperbolic, satellite, or torus knots, and one important family is Lorenz links, or T-links, which arise from dynamics. However, it remains difficult to determine the geometric type of a Lorenz link from a description via dynamics or as a T-link. In this paper, we consider those T-links that are torus links. We show that T-links obtained by full twists along torus links can never be torus links, aside from a family of cases. This addresses a question of Birman and Kofman [J. Topol. 2 (2009), pp. 227–248].

Explicit formulae for Chern-Simons invariants of the hyperbolic J(2n,-2m) knot orbifolds

We calculate the Chern-Simons invariants of the hyperbolic double twist knot orbifolds using the Schläfli formula for the generalized Chern-Simons function on the family of cone-manifold structures of double twist knots.

Characterizing slopes for the -pretzel knot

AbstractIn this note, we exhibit concrete examples of characterizing slopes for the knot $12n242$ , also known as the $(-2,3,7)$ -pretzel knot. Although it was shown by Lackenby that every knot admits infinitely many characterizing slopes, the nonconstructive nature of the proof means that there are very few hyperbolic knots for which explicit examples of characterizing slopes are known.

Slow-fast torus knots

The aim of this paper is to study global dynamics of $C^\infty$-smooth slow-fast systems on the $2$-torus of class $C^\infty$ using geometric singular perturbation theory and the notion of slow divergence integral. Given any $m\in\mathbb{N}$ and two relatively prime integers $k$ and $l$, we show that there exists a slow-fast system $Y_{\epsilon}$ on the $2$-torus that has a $2m$-link of type $(k,l)$, i.e. a (disjoint finite) union of $2m$ slow-fast limit cycles each of $(k,l)$-torus knot type, for all small $\epsilon>0$. The $(k,l)$-torus knot turns around the $2$-torus $k$ times meridionally and $l$ times longitudinally. There are exactly $m$ repelling limit cycles and $m$ attracting limit cycles. Our analysis: a) proves the case of normally hyperbolic singular knots, and b) provides sufficient evidence to conjecture a similar result in some cases where the singular knots have regular nilpotent contact with the fast foliation.

Bounds on Pachner moves and systoles of cusped 3–manifolds

Any two geometric ideal triangulations of a cusped complete hyperbolic $3$-manifold $M$ are related by a sequence of Pachner moves through topological triangulations. We give a bound on the length of this sequence in terms of the total number of tetrahedra and a lower bound on dihedral angles. This leads to a naive but effective algorithm to check if two hyperbolic knots are equivalent, given geometric ideal triangulations of their complements. Given a geometric ideal triangulation of $M$, we also give a lower bound on the systole length of $M$ in terms of the number of tetrahedra and a lower bound on dihedral angles.

Hidden symmetries of hyperbolic links

W. D. Neumann and A. W. Reid conjectured that the figure-eight knot and the two dodecahedral knots are the only hyperbolic knots admitting hidden symmetries. We construct an $n$-component hyperbolic link whose complement admits hidden symmetries for any $n\ge 4$.