arXiv research

Study of cable links of uniformly thick knots, revealing new isotopy phenomena.

New virtual braid group structure and presentation found.

New proof shows most thin knots satisfy Cabling Conjecture.

Formula derived for cabled knots' concordance invariants.

New series invariant for knots and cables, with robustness and relations.

New invariant detects infinite order cabled knots.

New insights into knot fusion numbers via cabling.

New phenomena in knot thickness revealed by Legendrian large cables.

The paper classifies Legendrian and transverse knots in cable knot types.

Study links with annuli using sutured Floer homology.

Study shows crossing numbers of cable knots are larger than previously thought.

Estimates heat kernel gradients on fractal-like cable systems.

A new method calculates the minimum volume swept by a sphere's homotopy in 3D space.

The paper generalizes BPS-series for $(2,2w+1)$-cabling of figure eight knot.

Knots' Morse-Novikov number behaves additively under connected sum and unchanged by cabling.

Lower bounds on unknotting number for cabled knots.

We study the incompressible surfaces in the exterior of a cable knot and use this to compute the representativity and waist of most cable knots.

Study shows $(p,q)$-cables of non-trivial knots are not thin.

We continue our study of the degree of the colored Jones polynomial under knot cabling started in "Knot Cabling and the Degree of the Colored Jones Polynomial" (arXiv:1501.01574). Under certain hypothesis on this degree, we determine how the Jones slopes and the linear term behave under cabling. As an application we ve…

Perturbing non-minimal bridge positions of a knot ensures similar behavior for its cable links.

We prove a cabling formula for the concordance invariant $ν^+$, defined by the author and Hom. This gives rise to a simple and effective 4-ball genus bound for many cable knots.

New polynomial criterion for periodic knots identified.

We continue our study of the knot Floer homology invariants of cable knots. For large |n|, we prove that many of the filtered subcomplexes in the knot Floer homology filtration associated to the (p,pn+1) cable of a knot, K, are isomorphic to those of K. This result allows us to obtain information about the behavior of …

Study tunnel numbers of cable knots and their companions, proving new bounds and constructing examples.

Let $k\subset S^3$ be a nontrivial knot. The Cabling Conjecture of Francisco González-Acuña and Hamish Short posits that $π$-Dehn surgery on $k$ produces a reducible manifold if and only if $k$ is a $(p,q)$-cable knot and the surgery slope $π$ equals $pq$. We extend the work of James Allen Hoffman to prove the Cabling …

Classifies surgeries on torus knots and cables that bound rational homology balls.

We prove that the class of topological knot types that are both Legendrian simple and satisfy the uniform thickness property (UTP) is closed under cabling. An immediate application is that all iterated cabling knot types that begin with negative torus knots are Legendrian simple. We also examine, for arbitrary numbers …

Smooth figure-eight knot cables have infinite order.

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

System classifies metaphorical violence on cable news.

We study the behavior of the degree of the colored Jones polynomial and the boundary slopes of knots under the operation of cabling. We show that, under certain hypothesis on this degree, if a knot $K$ satisfies the Slope Conjecture then a $(p, q)$-cable of $K$ satisfies the conjecture, provided that $p/q$ is not a Jon…

Formula calculates knot Floer complexes for specific cable knots.

We define a concordance invariant, epsilon(K), associated to the knot Floer complex of K, and give a formula for the Ozsváth-Szabó concordance invariant tau of K_{p,q}, the (p,q)-cable of a knot K, in terms of p, q, tau(K), and epsilon(K). We also describe the behavior of epsilon under cabling, allowing one to compute …

Cables of L-space knots have multiplicative knot Floer order.

We study the AJ conjecture for $(r,2)$-cables of a knot, where $r$ is an odd integer. Using skein theory, we show that the AJ conjecture holds true for most $(r,2)$-cables of some classes of two-bridge knots and pretzel knots.

We show that most cabled knots over torus knots in $S^3$ satisfy the AJ-conjecture, namely each $(r,s)$-cabled knot over each $(p,q)$-torus knot satisfies the $AJ$-conjecture if $r$ is not a number between $0$ and $pqs$.

We prove an explicit cabling formula for the colored Jones polynomial. As an application we prove the volume conjecture for all zero volume knots and links, i.e. all knots and links that are obtained from the unknot by repeated cabling and connected sum.

Cabling explained using curves on a punctured torus.

Develops slope detection for 3-manifolds with torus boundaries.

We show that most cabled knots over the figure eight knot in $S^3$ satisfy the AJ-conjecture, in particular, any $(r,s)$-cabled knot over the figure eight knot satisfies the $AJ$-conjecture if $r$ is not a number between $-4s$ and $4s$.

In this note we study Legendrian and transverse knots in the knot type of a (p,q)-cable of a knot K in 3-sphere. We give two structural theorems that describe when the (p,q)-cable of a Legendrian simple knot type K is also Legendrian simple.

We give a formula of the Upsilon invariant of any L-space cable knot $K_{p,q}$ using $p,Υ_K$ and $Υ_{T_{p,q}}$. The integral value of the Upsilon invariant gives a ${\mathbb Q}$-valued knot concordance invariant. We compute the integral values for L-space iterated cable knots.

We establish the volume conjecture for (m,2)-cables of the figure 8 knot, when m is odd. For (m,2)-cables of general knots where m is even, we show that the limit in the volume conjecture depends on the parity of the color (of the Kashaev invariant). There are many cases when the volume conjecture for cables of the fig…

We study the behavior of the Ozsvath-Szabo and Rasmussen knot concordance invariants tau and s on K(m,n), the (m,n)-cable of a knot K where m and n are relatively prime. We show that for every knot K and for any fixed positive integer m, both of the invariants evaluated on K(m,n) differ from their value on the torus kn…

Study how Turaev-Viro invariants change with cabling operations.

This paper is devoted to the study of the knot Floer homology groups HFK(S^3,K_{2,n}), where K_{2,n} denotes the (2,n) cable of an arbitrary knot, K. It is shown that for sufficiently large |n|, the Floer homology of the cabled knot depends only on the filtered chain homotopy type of CFK(K). A precise formula for this …

The Strong Slope Conjecture is proven for specific knot types.

Deep learning boosts cable capacity by 19%.