Related papers: Proof of the volume conjecture for torus knots
We prove that any integer power of $2$ can be realized as the Tamagawa number of a torus attached to a CM algebra considered by Guo Sheu Yu and Liang Yang Yu. Such a result is obtained by Liang Yang Yu under assuming "a generalized Landau…
The purpose of this note is to explain that the combinatorial local log-concavity conjecture introduced by Gross, Mansour, Tucker and Wang (Eur. J. Comb. 52, 207-222, 2016) in fact follows from a result of Stanley (Eur. J. Comb. 32 (6),…
Given knots K and J, one can ask whether a single smoothing of a crossing in a diagram for K can convert it into a diagram for J. As an interesting example, Zekovic discovered that the torus knot T(2,5) can be converted into T(2,-5) with a…
Recently, it has been proven that a tensegrity framework that arises from coning the one-skeleton of a convex polytope is rigid. Since such frameworks are not always infinitesimally rigid, this leaves open the question as to whether they…
We study the alternating subspace of holomorphic sections of a special prequantum line bundle over SU(2)-character variety of torus, and show that it is isomorphic to the projective representation of mapping class group of peripheral torus…
In this paper, we prove Mahler's conjecture concerning the volume product of centrally symmetric convex bodies in $\mathbb{R}^n$ in the case where $n=3$. Furthermore, we determine the equality condition.
The trunk of a knot in $S^3$, defined by Makoto Ozawa, is a measure of geometric complexity similar to the bridge number or width of a knot. We prove that for any two knots $K_1$ and $K_2$, we have $tr(K_1 \# K_2) =…
Two conjectures recently proposed by one of the authors are disproved
The author proves that the generalized Suita conjecture holds for any complex torus, which means that $ \alpha\pi K \geq c^2(\alpha\in\mathbb R)$, $c$ being the modified logarithmic capacity and $K$ being the Bergman kernel on the diagonal.…
We show that for a large class of hyperbolic knots and links, we can determine bounds on the volume of the link complement from combinatorial information given by a link diagram. Specifically, there is a universal constant C such that if a…
The Vol-Det Conjecture relates the volume and the determinant of a hyperbolic alternating link in $S^3$. We use exact computations of Mahler measures of two-variable polynomials to prove the Vol-Det Conjecture for many infinite families of…
Twisted torus knots are a generalization of torus knots, obtained by introducing additional full twists to adjacent strands of the torus knots. In this article, we present an explicit formula for the Alexander polynomial of twisted torus…
Extending upon our previous work, we verify the Jones Unknot Conjecture for all knots up to $24$ crossings. We describe the method of our approach and analyze the growth of the computational complexity of its different components.
Using Kauffman's model of flat knotted ribbons, we demonstrate how all regular polygons of at least seven sides can be realised by ribbon constructions of torus knots. We calculate length to width ratios for these constructions thereby…
Twist tori are examples of exotic monotone lagrangian tori, presented in [1]. This tree of examples grew up over the first one --- the torus $\Theta \in \R^4$, constructured in [2] and [3]. On the other hand, in [4] and [5] we proposed a…
We outline an approach to prove the two dimensional Jacobian Conjecture using the theory of fractals.
The classical honeycomb conjecture asserts that any partition of the plane into regions of equal area has perimeter at least that of the regular hexagonal honeycomb tiling. Pappus discusses this problem in his preface to Book V. This paper…
In a recent talk of Robbert Fokkink, some conjectures related to the infinite Tribonacci word were stated by the speaker and the audience. In this note we show how to prove (or disprove) the claims easily in a "purely mechanical" fashion,…
We propose a class of toric Lagrangian A-branes on the resolved conifold that is suitable to describe torus knots on S^3. The key role is played by the SL(2,Z) transformation, which generates a general torus knot from the unknot. Applying…
The Slope Conjecture proposed by Garoufalidis asserts that the degree of the colored Jones polynomial determines a boundary slope, and its refinement, the Strong Slope Conjecture proposed by Kalfagianni and Tran asserts that the linear term…