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Soheil Azarpendar

Publications and source records attributed to Soheil Azarpendar.

7 recordsLinked to original sources
arXiv2025

Twisting, Stabilization and Bordered Floer homology

Consider an unknot $c$ in $S^3$ and a knot $K$ in ${S^3-N(c)}$. Twisting the knot $K$ along $c$, or equivalently applying $\frac{1}{m}$-surgery on $c$, produces a family of knots $\{K_m\}_{m \in \mathbb{Z}}$. We use bordered Floer homology and the theory of immersed curve invariants to show that for $|m|\gg0$, total dimension of $\widehat{\mathrm{HFK}}(K_m)$, $\tau(K_{m})$ and thickness of $K_{m}$ are linear functions of $m$. Furthermore, we prove that the extremal coefficients of the Alexander polynomial and extremal knot Floer homologies of $K_m$ stabilize as $m$ goes to infinity. This generalizes results of Chen, Lambert-Cole, Roberts, Van Cott and the author on coherent twist families.

math.GT↗
arXiv2025

Chainmail links, Dehn surgery number and $10/8$

Liu and Piccirillo developed a combinatorial argument that employs the 10/8-theorem to demonstrate that certain manifolds cannot be obtained via Dehn surgery on a knot. We extend their approach by creating additional examples using chainmail links.

math.GT↗
arXiv2024

On Fox's trapezoidal conjecture

We investigate Fox's trapezoidal conjecture for alternating links. We show that it holds for diagrammatic Murasugi sums of special alternating links, where all sums involved have length less than three (which includes diagrammatic plumbing). It also holds for links containing a large twist region, which we call twist-concentrated. Furthermore, we show some weaker inequalities between consecutive coefficients of the Alexander polynomial of an alternating 3-braid closure, and extend this to arbitrary alternating links. We then study an extension of the trapezoidal conjecture due to Hirasawa and Murasugi, which states that the stable length of the Alexander polynomial of an alternating link can be bounded from above using the signature. We estabilish this and determine when equality holds for diagrammatic Murasugi sums of special alternating knots where each sum has length less than three, and also for twist-concentrated 3-braids. Finally, we study the behavior of the Hirasawa-Murasugi inequality under concordance.

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arXiv2023

Negative definite spin filling and branched double covers

We investigate the negative definite spin fillings of branched double covers of alternating knots. We derive some obstructions for the existence of such fillings and find a characterization of special alternating knots based on them.

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arXiv2020

Lower bounds for the chromatic number of certain Kneser-type hypergraphs

Let $n\ge 1$, $r\ge 2$, and $s\ge 0$ be integers and ${\cal P}=\{P_1,\dots, P_l\}$ be a partition of $[n]=\{1,\dots, n\}$ with $|P_i|\le r$ for $i=1,\dots, l$. Also, let $\cal F$ be a family of non-empty subsets of $[n]$. The $r$-uniform Kneser-type hypergraph $\mbox{KG}^r({\cal F}, {\cal P},s)$ is the hypergraph with the vertex set of all $\cal P$-admissible elements $F\in {\cal F}$, that is $|F\cap P_i|\le 1$ for $i=1,\dots, l$ and the edge set of all $r$-subsets $\{F_1,\dots, F_r\}$ of the vertex set that $|F_i\cap F_j|\le s$ for all $1\le i<j\le r$. In this article, we extend the equitable $r$-colorability defect $\mbox{ecd}^r({\cal F})$ of Abyazi Sani and Alishahi to the case when one allows intersection among the vertices of an edge. It will be denoted by $\mbox{ecd}^r({\cal F},s)$. We then, give (under certain assumptions) lower bounds for the chromatic number of $\mbox{KG}^r({\cal F}, {\cal P},s)$ and some of its variants in terms of $\mbox{ecd}^r({\cal F},\lfloor s/2\rfloor)$. This work generalizes many existing results in the literature of the Kneser hypergraphs. It generalizes the previous results of the current authors from the special family of all $k$-subsets of $[n]$ to a general family $\cal F$ of subsets.

math.CO↗
arXiv2020

On some topological and combinatorial lower bounds on chromatic number of Kneser type hyper graphs

In this paper, we prove a generalization of a conjecture of Erdös, about the chromatic number of certain Kneser-type hypergraphs. For integers $n,k,r,s$ with $n\ge rk$ and $2\le s\le r$, the $r$-uniform general Kneser hypergraph $\mbox{KG}^r_s(n,k)$, has all $k$-subsets of $\{1,\dots,n\}$ as the vertex set and all multi-sets $\{A_1,\dots, A_r\}$ of $k$-subsets with $s$-wise empty intersections as the edge set. The case $r=s=2$, was considers by Kneser \cite{K} in 1955, where he conjectured that its chromatic number is $n-2(k-1)$. This was finally proved by Lovász \cite{L} in 1978. The case $r>2$ and $s=2$, was considered by Erdös in 1973, and he conjectured that its chromatic number is $\left\lceil\frac{n-r(k-1)}{r-1}\right\rceil$. This conjecture was proved by Alon, Frankl and Lovász \cite{AFL} in 1986. The case where $s>2$, was considered by Sarkaria \cite{S} in 1990, where he claimed to prove a lower bound for its chromatic number which generalized all previous results. Unfortunately, an error was found by Lange and Ziegler \cite{Z'} in 2006 in the induction method of Sarkaria on the number of prime factors of $r$, and Sarkaria's proof only worked when $s$ is less than the smallest prime factor of $r$ or $s=2$. In this paper, by applying the $\mathbb Z_p$-Tucker lemma of Ziegler \cite{Z} and Meunier \cite{M}, we finally prove the general Erdös conjecture and prove the claimed result of Sarkaria for any $2\le s\le r$. We also provide another proof of a special case of this result, using methods similar to those of Alon, Frankl, and Lovász \cite{AFL} and compute the connectivity of certain simplicial complexes that might be of interest in their own right.

math.CO↗