arXiv research

The paper proves limitations on hyperbolic knot complements with hidden symmetries.

We show that a hyperbolic 2-bridge knot complement is the unique knot complement in its commensurability class. We also discuss constructions of commensurable hyperbolic knot complements and put forth a conjecture on the number of hyperbolic knot complements in a commensurability class.

In this paper we analyze symmetries, hidden symmetries, and commensurability classes of $(ε, d_L)$-twisted knot complements, which are the complements of knots that have a sufficiently large number of twists in each of their twist regions. These knot complements can be constructed via long Dehn fillings on fully augmen…

Flat fully augmented links with homeomorphic complements are equivalent.

The complement of a non-separating planar graph contains a K_n minor.

Upper and lower bounds for hyperbolic rod complements in 3-torus volumes.

Knots in circle bundles are uniquely identified by their complements.

We show that all two-bridge knot and link complements are virtually fibered. We also show that spherical Montesinos knot and link complements are virtually fibered. This is accomplished by showing that such knot complements are finitely covered by great circle link complements.

In this paper, we show that any non-arithmetic hyperbolic $2$-bridge link complement admits no hidden symmetries. As a corollary, we conclude that a hyperbolic $2$-bridge link complement cannot irregularly cover a hyperbolic $3$-manifold. By combining this corollary with the work of Boileau and Weidmann, we obtain a ch…

This paper exhibits an infinite family of hyperbolic knot complements that have three knot complements in their respective commensurability classes.

Aitchison and Rubinstein constructed two knot complements that can be decomposed into two regular ideal dodecahedra. This paper shows that these knot complements are the only knot complements that decompose into n regular ideal dodecahedra, providing a partial solution to a conjecture of Neumann and Reid.

The study examines the topology of complements of polytopal skeletons.

Formula connects knot complements' invariants.

We give examples of non-fibered hyperbolic knot complements in homology spheres that are not commensurable to fibered knot complements in homology spheres. In fact, we give many examples of knot complements in homology spheres with the property that every commensurable knot complement in a homology sphere has non-monic…

Study on colored Jones polynomial and link complements.

Study Lagrangian Floer theory in smooth divisor complements.

New bounds on Heegaard distance for link complements in 3D space.

Paper computes hyperbolic structure of Borromean rings complement.

Preserves hyperbolicity in link complements with two moves.

The purpose of this note is to announce complete answers to the following questions. (1) For an essential simple loop on a 2-bridge sphere in a 2-bridge link complement, when is it null-homotopic in the link complement? (2) For two distinct essential simple loops on a 2-bridge sphere in a 2-bridge link complement, when…

New link types derived from old links.

We show that every hyperbolic link complement contains closed quasi-Fuchsian surfaces. As a consequence, we obtain the result that on a hyperbolic link complement, if we remove from each cusp of the manifold a certain finite set of slopes, then all remaining Dehn fillings on the link complement yield manifolds with clo…

We prove that there are compact submanifolds of the 3-sphere whose interiors are not homeomorphic to any geometric limit of hyperbolic knot complements.

Study on the existence of Dirac complements for Dirac structures.

For a given embedded Lagrangian in the complement of a complex hypersurface we show existence of a holomorphic disc in the complement having boundary on that Lagrangian.

Complete classification of rod complements in 3-torus using topology.

The study counts ideal points in 2-bridge knot complements using knot diagrams.

This paper completes a classification of the types of orientable and non-orientable cusps that can arise in the quotients of hyperbolic knot complements. In particular, $S^2(2,4,4)$ cannot be the cusp cross-section of any orbifold quotient of a hyperbolic knot complement. Furthermore, if a knot complement covers an orb…

We establish a pair of criteria for proving that most knot complements obtained as Dehn fillings of a given two-component hyperbolic link complement lack hidden symmetries. To do this, we use certain rational functions on varieties associated to the link. We apply our criteria to show that among certain infinite famili…

We consider the following question: when is the manifold obtained by gluing together two knot complements an $L$-space? Hedden and Levine proved that splicing 0-framed complements of nontrivial knots never produces an $L$-space. We extend this result to allow for arbitrary integer framings. We find that splicing two in…

We investigate commensurability classes of hyperbolic knot complements in the generic case of knots without hidden symmetries. We show that such knot complements which are commensurable are cyclically commensurable, and that there are at most $3$ hyperbolic knot complements in a cyclic commensurability class. Moreover …

We define the higher-order Alexander modules $A_{n,i}(\mathcal{U})$ and higher-order degrees $δ_{n,i}(\mathcal{U})$ which are invariants of a complex hypersurface complement $\mathcal{U}$. These invariants come from the module structure of the homology of certain solvable covers of the hypersurface complement. Such inv…

This paper examines number theoretic and topological properties of fully augmented pretzel link complements. In particular, we determine exactly when these link complements are arithmetic and exactly which are commensurable with one another. We show these link complements realize infinitely many CM-fields as invariant …

Study counts geodesic surfaces in knot complements, finding unique ones for small knots.

Study Weinstein structures on toric divisors' complements.

This paper provides two obstructions to small knot complements in $S^3$ admitting hidden symmetries. The first obstruction is being cyclically commensurable with another knot complement. This result provides a partial answer to a conjecture of Boileau, Boyer, Cebanu, and Walsh. We also provide a second obstruction to a…

Graphs and their complements are intrinsically knotted.

Study shows arithmetic properties of specific hyperbolic Dehn fillings.

3-manifolds with hyperbolic handlebody complements are studied.

Let K be a hyperbolic (-2,3,n) pretzel knot and M = S^3 K its complement. For these knots, we verify a conjecture of Reid and Walsh: there are at most three knot complements in the commensurability class of M. Indeed, if n \neq 7, we show that M is the unique knot complement in its class. We include examples to illustr…

We prove that any complete hyperbolic 3--manifold with finitely generated fundamental group, with a single topological end, and which embeds into $\BS^3$ is the geometric limit of a sequence of hyperbolic knot complements in $\BS^3$. In particular, we derive the existence of hyperbolic knot complements which contain ba…

Defines coarse cohomology of space complements, proving new duality results.

Checkerboard surfaces in alternating link complements are used frequently to determine information about the link. However, when many crossings are added to a single twist region of a link diagram, the geometry of the link complement stabilizes (approaches a geometric limit), but a corresponding checkerboard surface in…

Null-homotopic knots in certain 3-manifolds are uniquely identified by their complements.

The paper identifies knots in specific lens spaces based on their complements.

Learning with a primary objective, such as softmax cross entropy for classification and sequence generation, has been the norm for training deep neural networks for years. Although being a widely-adopted approach, using cross entropy as the primary objective exploits mostly the information from the ground-truth class f…

We construct an algorithm that lists all closed essential surfaces in the complement of a knot that lies on the fiber of a trefoil or figure eight knot. Such knots are Berge knots and hence admit lens space surgeries. Furthermore they may have arbitrarily large hyperbolic volume. Using this algorithm we concoct large v…

The study finds conditions on graph complements for positive curvature.