arXiv research

The study confirms a conjecture for a specific type of knot.

The Slope Conjecture is verified for a specific family of Montesinos knots.

Proves conjectures for pretzel knots using polynomial degrees.

The Strong Slope Conjecture is proven for specific knot types.

The slope conjecture proposed by Garoufalidis asserts that the Jones slopes given by the sequence of degrees of the colored Jones polynomials are boundary slopes. We verify the slope conjecture for graph knots, i.e. knots whose Gromov volume vanish.

Verify knot polynomial conjectures for specific knots.

The slope conjecture gives a precise relation between the degree of the colored Jones polynomial of a knot and the boundary slopes of essential surfaces in the knot complement. In this note we propose a generalization of the slope conjecture to links. We prove the conjecture for all alternating and more generally adequ…

Study Mazur doubles of knots and their relation to the Slope Conjecture.

Proving a conjecture for a specific type of knots.

The strong slope conjecture helps identify torus knots.

Paper tackles $L$-space conjecture for knot manifolds, proving equivalence for some properties.

Every knot has infinitely many characterising slopes.

Garoufalidis conjectured a relation between the boundary slopes of a knot and its colored Jones polynomials. According to the conjecture, certain boundary slopes are detected by the sequence of degrees of the colored Jones polynomials. We verify this conjecture for adequate knots, a class that vastly generalizes that o…

The paper proves a slope equality for smooth plane curve fibrations and applies it to Durfee's conjecture.

The study confirms conjectures about slopes of knots using knot Floer homology.

Study slopes in 3-manifolds, proving conjectures about knots.

Jones slopes detect figure eight knot, and characterize alternating knots.

The Slope Conjecture relates a quantum knot invariant, (the degree of the colored Jones polynomial of a knot) with a classical one (boundary slopes of incompressible surfaces in the knot complement). The degree of the colored Jones polynomial can be computed by a suitable (almost tight) state sum and the solution of a …

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…

Algorithm decides if knots satisfy a conjecture, linking Jones polynomial to surface complexity.

Characterizes slopes for Markov ordering on prime pairs.

The paper introduces Slope Conjecture which relates the degree of the Jones polynomial of a knot and its parallels with the slopes of incompressible surfaces in the knot complement. More precisely, we introduce two knot invariants, the Jones slopes (a finite set of rational numbers) and the Jones period (a natural numb…

Study slopes on knot manifolds to understand their fundamental groups.

Study knot invariants to deduce Hopf invariant and propose a slope conjecture.

New foliations show knot meridians are detectable.

We establish a characterization of adequate knots in terms of the degree of their colored Jones polynomial. We show that, assuming the Strong Slope conjecture, our characterization can be reformulated in terms of "Jones slopes" of knots and the essential surfaces that realize the slopes .For alternating knots the refor…

The paper studies slopes for knot fillings with left-orderable fundamental groups.

The study bounds exceptional surgeries for hyperbolic knots.

Study tangle equations linking enzyme actions to knot theory.

Which slopes can or cannot appear as Seifert fibered slopes for hyperbolic knots in the 3-sphere S^3? It is conjectured that if r-surgery on a hyperbolic knot in S^3 yields a Seifert fiber space, then r is an integer. We show that for each integer n, there exists a tunnel number one, hyperbolic knot K_n in S^3 such tha…

Khovanov homology detects essential surfaces in knot complements.

Dehn filling on $v2503$ creates non-orderable spaces.

This note gives the first example of a hyperbolic knot in the 3-sphere that lacks a nonorientable essential spanning surface; this disproves the Strong Neuwirth Conjecture formulated by Ozawa and Rubinstein. Moreover, this knot has no even strict boundary slopes, disproving the Even Boundary Slope Conjecture of the sam…

We prove that for 2-bridge knots, the diameter, D, of the set of boundary slopes is twice the crossing number, c. This constitutes partial verification of a conjecture that, for all knots in S^3, D is at most 2c.

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…

Torus decomposition shows foliation detected slopes for glued knot manifolds.

We give a simple criterion for slope stability of Fano manifolds $X$ along divisors or smooth subvarieties. As an application, we show that $X$ is slope stable along an ample effective divisor $D\subset X$ unless $X$ is isomorphic to a projective space and $D$ is a hyperplane section. We also give counterexamples to Au…

Detect slopes in toroidal 3-manifolds to prove properties of fundamental groups.

Geodesic rays prove key aspects of cscK metrics existence and stability.

Using work of Ozsvath and Szabo, we show that if a nontrivial knot in S^3 admits a lens space surgery with slope p, then p <= 4g+3, where g is the genus of the knot. This is a close approximation to a bound conjectured by Goda and Teragaito.

Using elementary ideas from Tropical Geometry, we assign a a tropical curve to every $q$-holonomic sequence of rational functions. In particular, we assign a tropical curve to every knot which is determined by the Jones polynomial of the knot and its parallels. The topical curve explains the relation between the AJ Con…

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 …

The exceptional Dehn filling conjecture of the second author concerning the relationship between exceptional slopes $α, β$ on the boundary of a hyperbolic knot manifold $M$ has been verified in all cases other than small Seifert filling slopes. In this paper we verify it when $α$ is a small Seifert filling slope and $β…

Translation distances in fibered 3-manifolds with boundary are bounded and grow with complexity.

Jones slopes and volume of near-alternating links studied.

Survey on minimal penalty algorithms and slope heuristics.

The study calculates and analyzes alternating surgeries for various knots.

Disproves conjectures about shared surgeries for distinct knots.