arXiv research

Proof of Knot Entropy Conjecture for tube lattice polygons.

Study on knotting in very long polymer chains, finding Poisson distribution for prime knot types.

Study proves Hecke lifting conjecture for torus knots and verifies it for any framed knots.

New proof shows most thin knots satisfy Cabling Conjecture.

New satellite knots counter a conjecture about Lorenz knots.

New evidence refutes old conjectures about knot homology ranks, suggesting new congruences.

Verifies a conjecture for the figure eight knot.

Counterexamples found for volume entropy conjecture in hyperbolic 3-manifolds.

The AJ conjecture, formulated by Garoufalidis, relates the A-polynomial and the colored Jones polynomial of a knot in the 3-sphere. It has been confirmed for all torus knots, some classes of two-bridge knots and pretzel knots, and most cabled knots over torus knots. The strong AJ conjecture, formulated by Sikora, relat…

Proves cosmetic crossing conjecture for certain knots.

The (Strong) Slope Conjecture relates the degree of the colored Jones polynomial of a knot to certain essential surfaces in the knot complement. We verify the Slope Conjecture and the Strong Slope Conjecture for 3-string Montesinos knots satisfying certain conditions.

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 $pqs$.

The abstract discusses conjectures about virtual Legendrian knots and their relation to causality.

A well-known conjecture in knot theory says that the percentage of hyperbolic knots amongst all of the prime knots of $n$ or fewer crossings approaches $100$ as $n$ approaches infinity. In this paper, it is proved that this conjecture contradicts several other plausible conjectures, including the 120-year-old conjectur…

Neuwirth asked if any non-trivial knot in the 3-sphere can be embedded in a closed surface so that the complement of the surface is a connected essential surface for the knot complement. In this paper, we examine some variations on this question and prove it for all knots up to 11 crossings except for two examples. We …

The Besson-Courtois-Gallot theorem is proven for noncompact finite volume Riemannian manifolds. In particular, no bounded geometry assumptions are made. This proves the minimal entropy conjecture for nonuniform rank one lattices.

Two knot families meet cosmetic surgery conjecture.

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 $4s$.

Conjecture $\mathbb{Z}$ is a knot theoretical equivalent form of the Kervaire Conjecture. We say that a knot have property $\mathbb{Z}$ if it satisfies Conjecture $\mathbb{Z}$ for that specific knot. In this work, we show that alternating Montesinos knots with three tangles have property $\mathbb{Z}$. We also show that…

Proves volume conjecture for twist knots using complex analysis.

The AJ conjecture relates the A-polynomial and the colored Jones polynomial of a knot in the 3-sphere. It has been verified for some classes of knots, including all torus knots, most double twist knots, (-2,3,6n \pm 1)-pretzel knots, and most cabled knots over torus knots. In this paper we study the AJ conjecture for (…

A conjecture proposed by J. Tripp in 2002 states that the crossing number of any knot coincides with the canonical genus of its Whitehead double. In the meantime, it has been established that this conjecture is true for a large class of alternating knots including $(2, n)$ torus knots, $2$-bridge knots, algebraic alter…

Proves ropelength conjecture for alternating knots.

It is conjectured that for each knot $K$ in $S^3$, the fundamental group of its complement surjects onto only finitely many distinct knot groups. Applying character variety theory we obtain an affirmative solution of the conjecture for a class of small knots that includes 2-bridge knots.

Disproves conjectures about shared surgeries for distinct knots.

In 1999, Kauffman-Harary conjectured that every non-trivial Fox $p$-coloring of a reduced, alternating knot diagram with prime determinant $p$ is heterogeneous. Ten years later this conjecture was proved by W. Mattman and P. Solis. Mathew Williamson generalized this conjecture to alternating virtual knots and proved it…

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

Verifies knot conjecture for 24-crossing knots.

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…

Proves volume conjectures for figure-eight knot surgeries.

Paper proves knots satisfy a conjecture using Jones polynomial.

We confirm the AJ conjecture [Ga04] that relates the A-polynomial and the colored Jones polynomial for those hyperbolic knots satisfying certain conditions. In particular, we show that the conjecture holds true for some classes of two-bridge knots and pretzel knots. This extends the result of the first author in [Le06]…

The cosmetic crossing conjecture (also known as the "nugatory crossing conjecture") asserts that the only crossing changes that preserve the oriented isotopy class of a knot in the 3-sphere are nugatory. We use the Dehn surgery characterization of the unknot to prove this conjecture for knots in integer homology sphere…

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.

Log-concave coefficient sequences for two-bridge knots proved.

Counterexamples found for knot conjectures.

We propose to generalize the volume conjecture to knotted trivalent graphs and we prove the conjecture for all augmented knotted trivalent graphs. As a corollary we find that for any link L there is a link containing L for which the volume conjecture holds.

New examples contradict a conjecture about knot surgeries.

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

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…

Proves special alternating knots cannot be decomposed as non-trivial band sums.

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.

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

The $AJ$-conjecture for a knot $K \subset S^3$ relates the $A$-polynomial and the colored Jones polynomial of $K$. If a two-bridge knot $K$ satisfies the $AJ$-conjecture, we give sufficient conditions on $K$ for the $(r,2)$-cable knot $C$ to also satisfy the $AJ$-conjecture. If a reduced alternating diagram of $K$ has …

Partial proof of a conjecture about knot concordance maps.

It is known that the fundamental group homomorphism $π_1(T^2) \to π_1(S^3\setminus K)$ induced by the inclusion of the boundary torus into the complement of a knot $K$ in $S^3$ is a complete knot invariant. Many classical invariants of knots arise from the natural (restriction) map induced by the above homomorphism on …

Constructs infinite families of hyperbolic knots satisfying a volume conjecture.

The paper conjectures Khovanov homology can distinguish torus and twist knots.