Related papers: Generalized Fishburn numbers and torus knots
Torus knots are an important family of knots about which much is understood; invariants of torus knots often exhibit nice formulas, making them convenient and fundamental building blocks for examples in knot theory. Spiral knots, defined…
Using techniques due to Coster, we prove a supercongruence for a generalization of the Domb numbers. This extends a recent result of Chan, Cooper and Sica and confirms a conjectural supercongruence for numbers which are coefficients in one…
We propose an algebraic model of the conjectural triply graded homology of Gukov, Dunfield and Rasmussen for some torus knots. It turns out to be related to the q,t-Catalan numbers of Garsia and Haiman.
We study certain linear representations of the knot group that induce augmentations of knot contact homology. This perspective on augmentations enhances our understanding of the relationship between the augmentation polynomial and the…
Let $L$ be a finite distributive lattice and $\mu : L \to {\mathbb R}^{+}$ a log-supermodular function. For functions $k: L \to {\mathbb R}^{+}$ let $$E_{\mu} (k; q) \defeq \sum_{x\in L} k(x) \mu (x) q^{{\mathrm rank}(x)} \in {\mathbb…
We estimate the covariance in counts of almost-primes in $\mathbb{F}_q[T]$, weighted by higher-order von Mangoldt functions. The answer takes a pleasant algebraic form. This generalizes recent work of Keating and Rudnick that estimates the…
We show that Hilbert schemes of planar curve singularities and their parabolic variants can be interpreted as certain generalized affine Springer fibers for $GL_n$, as defined by Goresky-Kottwitz-MacPherson. Using a generalization of affine…
We introduce a family of generalized Schr\"oder polynomials $S_\tau(q,t,a)$, indexed by triangular partitions $\tau$ and prove that $S_\tau(q,t,a)$ agrees with the Poincar\'e series of the triply graded Khovanov-Rozansky homology of the…
In this paper we propose a quantum gauge system from which we construct generalized Wilson loops which will be as quantum knots. From quantum knots we give a classification table of knots where knots are one-to-one assigned with an integer…
In this paper, we present several new congruences on the $q$-trinomial coefficients introduced by Andrews and Baxter. A new congruence on sums of central $q$-binomial coefficients is also established.
In this paper we will associate a family $\{K_1,\dots,K_l\}\subset \mathbb{S}^3$ of iterated torus knots to a given free numerical semigroup. We will describe the fundamental group of the knot complement of each knot of the family. Finally,…
Given an arbitrary ordered pair of coprime integers (a,b) we obtain a pair of identities of the Rogers--Ramanujan type. These identities have the same product side as the (first) Andrews--Gordon identity for modulus 2ab\pm 1, but an…
We show that two classes of combinatorial objects--inversion tables with no subsequence of decreasing consecutive numbers and matchings with no 2-nestings--are enumerated by the Fishburn numbers. In particular, we give a simple bijection…
\'{E}.\~Ghys proved that the linking numbers of modular knots and the "missing" trefoil $K_{2,3}$ in $S^3$ coincide with the values of a highly ubiquitous function called the Rademacher symbol for ${\rm SL}_2\mathbb{Z}$. In this paper, we…
In their 2004 paper, Stretched Littlewood-Richardson and Kostka Coefficients, King, Tollu, and Toumazet conjectured that if a Littlewood-Richardson coefficient of value 2 is stretched by a factor of N, the resulting coefficient has value…
We prove that the meridional rank and the bridge number of the Whitehead double of a prime algebraically tame knot coincide. Algebraically tame knots are a broad generalization of torus knots and iterated cable knots.
Transformation formulas for four-parameter refinements of the q-trinomial coefficients are proven. The iterative nature of these transformations allows for the easy derivation of several infinite series of q-trinomial identities, and can be…
The A-polynomial of a knot is defined in terms of SL(2,C) representations of the knot group, and encodes information about essential surfaces in the knot complement. In 2005, Dunfield-Garoufalidis and Boyer-Zhang proved that it detects the…
Using computational techniques we tabulate prime knots up to five crossings in the solid torus and the infinite family of lens spaces $L(p,q)$. For these knots we calculate the second and third skein module and establish which prime knots…
We show that the triple-crossing number of any knot is greater or equal to twice its (canonical) genus and we show an even stronger bound in the case of links. As an application we show that this bound is strong enough to obtain the…