Related papers: Pretzel links, mutation, and the slice-ribbon conj…
This paper is devoted to the study of the log-convexity of combinatorial sequences. We show that the log-convexity is preserved under componentwise sum, under binomial convolution, and by the linear transformations given by the matrices of…
We prove a prime decomposition theorem for string links in a thickened surface. Namely, we prove that any non-braid string link $\ell \subset \Sigma \times I$, where $\Sigma$ is a compact orientable (not necessarily closed) surface other…
We show that if $p/q$-surgery on a nontrivial knot $K$ yields the branched double cover of an alternating knot or link, then $|p/q|\leq 4g(K)+3$. This generalises a bound for lens space surgeries first established by Rasmussen. We also show…
The coefficients of the Kazhdan-Lusztig polynomials $P_{v,w}(q)$ are nonnegative integers that are upper semicontinuous on Bruhat order. Conjecturally, the same properties hold for $h$-polynomials $H_{v,w}(q)$ of local rings of Schubert…
We model the behaviour of a network of interacting (p,q) strings from IIB string theory by considering a field theory containing multiple species of string, allowing us to study the effect of non-intercommuting events due to two different…
Important objects of study in $\tau$-tilting theory include the $\tau$-tilting pairs over an algebra on the form $kQ/I$, with $kQ$ being a path algebra and $I$ an admissible ideal. In this paper, we study aspects of the combinatorics of…
Qazaqzeh and Chbili showed that for any quasi-alternating link, the degree of $Q$-polynomial is less than its determinant. We give a refinement of their evaluation.
The set of isotopy classes of nontrivial torus knots $T(p,q)$ in $S^3$ is in bijection with the set of coprime integer pairs $(p,q)$ satisfying $|p|>q\geq 2$. We verify the AJ conjecture for the connected sums $T(p,q)\# T(a,b)$ when $p$ and…
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…
Given a link L in the 3-sphere, we ask whether the components of L bound disjoint, nullhomologous disks properly embedded in a simply-connected positive-definite smooth 4-manifold; the knot case has been studied extensively in work of…
We study the eta-invariants of links and show that in many cases they form link concordance invariants, in particular that many eta-invariants vanish for slice links. This result contains and generalizes previous invariants by Smolinsky and…
We introduce the notion of a half-ribbon Hopf algebra, which is a ribbon Hopf algebra along with a distinguished element $t$ corresponding to twisting a ribbon by 180 degrees (the half-twist). We show that U_q(g) is a (topological)…
The material gives a new combinatorial proof of the multiplicative property of the S-transform. In particular, several properties of the coefficients of its inverse are connected to non-crossing linked partitions and planar trees.
A closer look at an example introduced by Livingston & Melvin and later studied by Miyazaki shows that a plumbing of two fibered ribbon knots (along their fiber surfaces) may be algebraically slice yet not ribbon.
For a finite lattice L, the congruence lattice Con L of L can be easily computed from the partially ordered set J(L) of join-irreducible elements of L and the join-dependency relation D\_L on J(L). We establish a similar version of this…
We show there exists a linear embedding of $K_{3,3,1}$ with n nontrivial 2-component links if and only if n = 1, 2, 3, 4, or 5.
The zig-zag conjecture says that the reductions of two-dimensional crystalline representations of the Galois group of ${\mathbb {Q}}_p$ of large exceptional weights and half-integral slopes up to $\frac{p-1}{2}$ vary through an alternating…
An n-dimensional \mu-component boundary link is a codimension 2 embedding of spheres L=\bigsqcup_{\mu}S^n \subset S^{n+2} such that there exist \mu disjoint oriented embedded (n+1)-manifolds which span the components of L. An F_\mu-link is…
Rudolph conjectures that for permutations $p$ and $q$ of the same length, $A_n(p) \le A_n(q)$ for all $n$ if and only if the spine structure of $T(p)$ is less than or equal to the spine structure of $T(q)$ in refinement order. We prove one…
We prove that a special alternating knot does not decompose as a non-trivial band sum. This restricts concordances from special alternating knots, and we conjecture that special alternating knots are ribbon concordance minimal. We verify…