arXiv research

We study the asymptotics of the higher dimensional Reidemeister torsion for torus knot exteriors, which is related to the results by W. Müller and P. Menal-Ferrer and J. Porti on the asymptotics of the Reidemeister torsion and the hyperbolic volumes for hyperbolic 3-manifolds. We show that the sequence of log |the high…

New families of twisted torus knots found with essential surfaces.

Study shows $(p,q)$-cable knots cannot undergo certain types of surgery.

Discretizes Hodge-Dirac operators on a torus.

Let $K= K(w,b,t)$ be a 1-bridge braid in a solid torus $V$, and let $γ$ be a $(p,q)$ curve on the torus $T = \partial V$ of the exterior $M_K$ of $K$. It will be shown that Dehn filling on $T$ along $γ$ produces a solid torus if and only if $p$ and $q$ satisfy one of four conditions determined by the parameters $(w,b,t…

Little is known on the classification of Heegaard splittings for hyperbolic 3-manifolds. Although Kobayashi gave a complete classification of Heegaard splittings for the exteriors of 2-bridge knots, our knowledge of other classes is extremely limited. In particular, there are very few hyperbolic manifolds that are know…

We study the way a strongly irreducible Heegaard surface $Σ$ intersects a knot exterior $X$ embedded in a 3-manifold, and show that if $Σ\cap \partial X$ consists of simple closed curves which are essential in both $Σ$ and $\partial X$, then the intersection $X \cap Σ$ consists of meridional annuli only. As an applicat…

Operational guide to discrete exterior calculus on cubic cells.

A knot K is called a 1-genus 1-bridge knot in a 3-manifold M if (M,K) has a Heegaard splitting (V_1,t_1)\cup (V_2,t_2) where V_i is a solid torus and t_i is a boundary parallel arc properly embedded in V_i. If the exterior of a knot has a genus 2 Heegaard splitting, we say that the knot has an unknotting tunnel. Natura…

A knot K in a closed connected orientable 3-manifold M is called a 1-genus 1-bridge knot if (M,K) has a splitting into two pairs of a solid torus V_i (i=1,2) and a boundary parallel arc in it. The splitting induces a genus two Heegaard splitting of the exterior of K naturally, i.e., K has an unknotting tunnel. However …

Algorithm finds knot diagrams from exterior triangulations.

The state of a knot is defined in the realm of Chern-Simons topological quantum field theory as a holomorphic section on the SU(2) character manifold of the peripheral torus. We compute the asymptotics of the torus knot states in terms of the Alexander polynomial, the Reidemeister torsion and the Chern-Simons invariant…

Supose that $Y$ is a lens space with $|H_1(Y; \mathbb{Z})|$ prime, and $Y$ does not contain a genus one fibered knot. We show that $Y$ contains a knot whose exterior is a once-punctured torus bundle if and only if $Y$ is the result of $p/q$-surgery on the trefoil. This partially answers a question posed by Ken Baker in…

Study shows torsion in knot module for specific Montesinos knots.

The paper supports a conjecture about a vanishing identity for certain 3-manifolds.

Computes knot groups for torus links using Seifert-van Kampen theorem.

Periods of parallel exterior forms define natural coordinates on the deformation space of complete affine structures on the two-torus. These coordinates define a differentiable structure on this deformation space, under which it is diffeomorphic to $R^2$. The action of the mapping class group of $T^2$ is equivalent in …

Suppose $M$ is a hyperbolic 3-manifold which admits two Dehn fillings $M(r_1)$ and $M(r_2)$ such that $M(r_1)$ contains an essential torus and $M(r_2)$ contains an essential annulus. It is known that $Δ= Δ(r_1, r_2) \leq 5$. We will show that if $Δ= 5$ then $M$ is the Whitehead sister link exterior, and if $Δ= 4$ then …

Parallelizes DEC on curved meshes using group actions.

We consider a numerical approach for the incompressible surface Navier-Stokes equation. The approach is based on the covariant form and uses discrete exterior calculus (DEC) in space and a semi-implicit discretization in time. The discretization is described in detail and related to finite difference schemes on stagger…

We determine the non-null homologous knots in lens spaces whose exteriors contain properly embedded once-punctured tori. All such knots arise as surgeries on the Whitehead link and are grid number 1 in their lens spaces. As a corollary, we classify once-punctured torus bundles that admit a lens space filling.

In this article, we give an explicit formula to compute the non-abelian twisted sign-determined Reidemeister torsion of the exterior of a fibered knot in terms of its monodromy. As an application, we give explicit formulae for the non abelian Reidemeister torsion of torus knots and of the figure eight knot.

In this paper we study exceptional Dehn fillings on hyperbolic knot manifolds which contain an essential once-punctured torus. Let $M$ be such a knot manifold and let $β$ be the boundary slope of such an essential once-punctured torus. We prove that if Dehn filling $M$ with slope $α$ produces a Seifert fibred manifold,…

Study shows Heegaard genus relation in 3-manifold amalgamation.

The paper extends torus surgery results in 4-manifolds and shows diffeomorphisms.

We determine the PSL_2(C) and SL_2(C) character varieties of the once-punctured torus bundles with tunnel number one, i.e. the once-punctured torus bundles that arise from filling one boundary component of the Whitehead link exterior. In particular, we determine `natural' models for these algebraic sets, identify them …

Framework for reconstructing knot exterior surface complexes.

Geometric finiteness theory for essential surfaces in knot exteriors with geometric bounds.

The study finds counterexamples to a conjecture about incompressible planar surfaces in hyperbolic link exteriors.

This paper classifies knots with simple curves in genus 2 handlebodies.

Suppose $K$ is a hyperbolic knot in a solid torus $V$ intersecting a meridian disk $D$ twice. We will show that if $K$ is not the Whitehead knot and the frontier of a regular neighborhood of $K \cup D$ is incompressible in the knot exterior, then $K$ admits at most one exceptional surgery, which must be toroidal. Embed…

Two de Rham complexes in diffeology are compared using a factor map.

New method calculates Thurston norm for 3-manifolds with toroidal boundaries.

The Euler class of taut foliations on Dehn fillings is studied, leading to new left-orderable slopes.

The paper proves a vanishing identity for twist knots using character varieties.

Let K be a tame knot with irreducible exterior M(K) in a closed, connected, orientable 3--manifold Sigma such that pi_1(Sigma) is cyclic. If infinity is not a strict boundary slope, then the diameter of the set of strict boundary slopes of K, denoted d_K, is a numerical invariant of K. We show that either (i) d_K >= 2 …

Let M be a compact connected orientable 3-manifold, with non-empty boundary that contains no 2-spheres. We investigate the existence of two properly embedded disjoint surfaces S_1 and S_2 such that M - (S_1 \cup S_2) is connected. We show that there exist two such surfaces if and only if M is neither a Z_2 homology sol…

Study exterior maximal surfaces with precise asymptotic behavior.

For the purposes of this paper, Dehn surgery along a curve K in a 3-manifold M with slope r is `exceptional' if the resulting 3-manifold M_K(r) is reducible or a solid torus, or the core of the surgery solid torus has finite order in the fundamental group of M_K(r). We show that, providing the exterior of K is irreduci…

A genus-1 tangle G is an arc properly embedded in a standardly embedded solid torus S in the 3-sphere. We say that a genus-1 tangle embeds in a knot K in S^3 if the tangle can be completed by adding an arc exterior to the solid torus to form the knot K. We call K a closure of G. An obstruction to embedding a genus-1 ta…

Classifies essential annuli in a genus two handlebody exterior.

We prove that every finitely presentable group G arises as the fundamental group of an orientable 3-complex obtained from a hyperbolic link complement, by coning each boundary torus of the link exterior to a distinct point. We define the closed-link-genus, clg(G), of any finitely presentable group G, which completely c…

We expect manifolds obtained by Dehn filling to inherit properties from the knot manifold. To what extent does that hold true for the Heegaard structure? We study four changes to the Heegaard structure that may occur after filling: (1) Heegaard genus decreases, (2) a new Heegaard surface is created, (3) a non-stabilize…

The paper classifies symmetries and determines exterior types of knotted handlebodies.

We give a topological characterisation of alternating knot exteriors based on the presence of special spanning surfaces. This shows that alternating is a topological property of the knot exterior and not just a property of diagrams, answering an old question of Fox. We also give a characterisation of alternating link e…

The study classifies area-maximizing hypersurfaces with singularities and exterior domains.

Discrete exterior calculus shows natural properties of wedge product and averaging.

The study finds infinite knot exteriors with meridional surfaces of any genus and boundary components.