arXiv research

We revisit the issue of the existence of infinitely many distinct prime knots with the same Alexander invariant. We present infinitely many distinct families, each family made up of infinitely many distinct knots. Within each family, the Alexander invariant is the same. Unlike other examples in the literature, ours are…

Paper constructs infinitely many non-braid positive hyperbolic L-space knots.

Study proves 0-surgery characterizes infinitely many knots.

Infinite knots have non-integer trace values.

New examples of knots with infinitely many inequivalent slice disks.

Study confirms infinitely many non-characterizing slopes for various knots.

Study shows infinite hyperbolic knots with odd torsion in Khovanov homology.

Classifies knot traces with specific trisection genus limits.

New research finds infinitely many hyperbolic knots not almost-fibered.

The paper finds infinite knots with surfaces of any positive genus.

Certain torus knots have infinitely many slopes that do not uniquely identify them.

New infinite families of twisted torus knots found.

New knots bound multiple non-isotopic ribbon disks.

We consider the existence of simple closed geodesics or "geodesic knots" in finite volume orientable hyperbolic 3-manifolds. Previous results show that at least one geodesic knot always exists [Bull. London Math. Soc. 31(1) (1999) 81-86], and that certain arithmetic manifolds contain infinitely many geodesic knots [J. …

Study finds infinite non-fibered twisted torus knots.

We say a knot $k$ in the 3-sphere $\mathbb S^3$ has {\it Property $IE$} if the infinite cyclic cover of the knot exterior embeds into $\mathbb S^3$. Clearly all fibred knots have Property $IE$. There are infinitely many non-fibred knots with Property $IE$ and infinitely many non-fibred knots without property $IE$. Both…

New invariant detects infinite order cabled knots.

The paper constructs infinitely many prime hyperbolic knots.

Although there are infinitely many knots with superbridge index n for every even integer n>2, there are only finitely many knots with superbridge index 3.

New method finds infinitely many knots in 3-manifolds.

This paper considers "geometric" ideal triangulations of cusped hyperbolic 3-manifolds, i.e. decompositions into positive volume ideal hyperbolic tetrahedra. We exhibit infinitely many geometric ideal triangulations of the figure eight knot complement. As far as we know, this is the first construction of infinitely man…

The group of any nontrivial torus knot, hyperbolic 2-bridge knot, or hyperbolic knot with unknotting number one contains infinitely many elements, none the automorphic image of another, such that each normally generates the group.

We show that a knot in $S^3$ with an infinite number of distinct incompressible Seifert surfaces contains a closed incompressible surface in its complement.

Twisting a knot $K$ in $S^3$ along a disjoint unknot $c$ produces a twist family of knots $\{K_n\}$ indexed by the integers. Comparing the behaviors of the Seifert genus $g(K_n)$ and the slice genus $g_4(K_n)$ under twistings, we prove that if $g(K_n) - g_4(K_n) < C$ for some constant $C$ for infinitely many integers $…

We investigate great circle links in the three-sphere, the class of links where each component is a great circle. Using the geometry of their complements, we classify such links up to five components. For any two-bridge knot complement, there is a finite cover that is the complement of a link of great circles in $S^3$.…

New findings show infinitely many knots cannot be smoothly round handle slices.

Links can be transformed into many others using a specific operation.

We prove that for any integer $n$ there exist infinitely many different knots in $S^3$ such that $n$-surgery on those knots yields the same 3-manifold. In particular, when $|n|=1$ homology spheres arise from these surgeries. This answers Problem 3.6(D) on the Kirby problem list. We construct two families of examples, t…

By proving a connected sum formula for the Legendrian invariant $λ_+$ in knot Floer homology we exhibit infinitely many transversely non simple knots.

Constructs infinitely many non-equivalent wild knots in Menger sponge.

The study finds infinite meridional essential surfaces in hyperbolic knot exteriors of various genera.

In the present note, we will show that there are infinitely many composite twisted torus knots.

The construction of knots via annular twisting has been used to create families of knots yielding the same manifold via Dehn surgery. Prior examples have all involved Dehn surgery where the surgery slope is an integral multiple of 2. In this note we prove that for any integer $n$ there exist infinitely many different k…

New knot homologies detect non-fibered knots, expanding on previous results.

We show that there are infinitely many pairs of alternating pretzel knots whose Jones polynomials are identical.

The study finds infinitely many twist knot complements with totally geodesic surfaces.

Any knot group is the image of the group of a prime knot by a homomorphism that preserves peripheral structure. In fact, there are infinitely many such prime knots. A related partial order on knots is defined, and its properties are discussed.

New hyperbolic knots with convex Upsilon invariants constructed.

New knots found with tough, unsliceable discs.

The Gordian distance between two knots measures how many crossing changes are needed to transform one knot into the other. It is known that there are always infinitely many non-equivalent knots `between' a pair of knots of Gordian distance two. In this paper we prove an extreme generalisation of this fact: there are kn…

A knot in the 3-sphere is called an L-space knot if it admits a nontrivial Dehn surgery yielding an L-space, i.e. a rational homology 3-sphere with the smallest possible Heegaard Floer homology. Given a knot K, take an unknotted circle c and twist K n times along c to obtain a twist family { K_n }. We give a sufficient…

We define an simple invariant of an embedded nullhomologous Lagrangian torus and use this invariant to show that many symplectic 4-manifolds have infinitely many pairwise symplectically inequivalent nullhomologous Lagrangian tori. We further show that for a large class of examples that lambda(T) is actually a C-infinit…

We define the virtual bridge number $vb(K)$ and the virtual unknotting number $vu(K)$ invariants for virtual knots. For ordinary knots $K$ they are closely related to the bridge number $b(K)$ and the unknotting number $u(K)$ and we have $vu(K)\leq u(K), vb(K)\leq b(K).$ There are no ordinary knots $K$ with $b(K)=1.$ We…

Alexander group systems for virtual long knots are defined and used to show that any virtual knot is the closure of infinitely many long virtual knots. Manturov's result that there exists a pair of long virtual knots that do not commute is reproved.

In this paper we construct some invariants of spatial graphs by disk-summing the constituent knots and show the delta edge-homotopy invariance of them. As an application, we show that there exist infinitely many slice spatial embeddings of a planar graph up to delta edge-homotopy, and there exist infinitely many bounda…

The paper constructs many knotted and linked objects in higher dimensions.

Satellite operators generate infinite rank subgroups in knot concordance.

We study knots in $S^3$ with infinitely many $SU(2)$-cyclic surgeries, which are Dehn surgeries such that every representation of the resulting fundamental group into $SU(2)$ has cyclic image. We show that for every such nontrivial knot $K$, its set of $SU(2)$-cyclic slopes is bounded and has a unique limit point, whic…