Abstract. In this paper, we investigate representations of links that are either centrally symmetric in [Formula: see text] or antipodally symmetric in [Formula: see text]. By using the notions of antipodally self-dual and antipodally symmetric maps, introduced and studied by the authors in [L. Montejano, J. L. Ramírez Alfonsín, and I. Rasskin, SIAM J. Discrete Math., 36 (2022), pp. 1551–1566], we are able to present sufficient combinatorial conditions for a link [Formula: see text] to admit such representations. The latter naturally provide sufficient conditions for [Formula: see text] to be amphichiral. We also introduce another (closely related) method yielding again sufficient conditions for [Formula: see text] to be amphichiral. We finally prove that a link [Formula: see text], associated to a map [Formula: see text], is amphichiral if the self-dual pairing of [Formula: see text] is not one of 6 specific cases among the classification of the 24 self-dual pairing [Formula: see text].

A self-dual map $G$ is said to be antipodally self-dual if the dual map $G^*$ is antipodal embedded in $\mathbb{S}^2$ with respect to $G$. In this paper, we investigate necessary and/or sufficient conditions for a map to be antipodally self-dual. In particular, we present a combinatorial characterization for map $G$ to be antipodally self-dual in terms of certain involutive labelings. The latter lead us to obtain necessary conditions for a map to be strongly involutive (a notion relevant for its connection with convex geometric problems). We also investigate the relation of antipodally self-dual maps and the notion of antipodally symmetric maps. It turns out that the latter is a very helpful tool to study questions concerning the symmetry as well as the amphicheirality of links.

Let $D$ be a knot diagram, and let ${\mathcal D}$ denote the set of diagrams that can be obtained from $D$ by crossing exchanges. If $D$ has $n$ crossings, then ${\mathcal D}$ consists of $2^n$ diagrams. A folklore argument shows that at least one of these $2^n$ diagrams is unknot, from which it follows that every diagram has finite unknotting number. It is easy to see that this argument can be used to show that actually ${\mathcal D}$ has more than one unknot diagram, but it cannot yield more than $4n$ unknot diagrams. We improve this linear bound to a superpolynomial bound by showing that at least $2^{\sqrt[3]{n}}$ of the diagrams in ${\mathcal D}$ are unknot. We also show that either all the diagrams in ${\mathcal D}$ are unknot or there is a diagram in ${\mathcal D}$ that is a diagram of the trefoil knot.

Let $k,d,\lambda \geqslant 1$ be integers with $d\geqslant \lambda $ and let $X$ be a finite set of points in $\mathbb{R}^{d}$. A $(d-\lambda)$-plane $L$ transversal to the convex hulls of all $k$-sets of $X$ is called a Kneser transversal. If in addition $L$ contains $(d-\lambda)+1$ points of $X$, then $L$ is called a complete Kneser transversal. In this paper, we present various results on the existence of (complete) Kneser transversals for $\lambda =2,3$. In order to do this, we introduce the notions of stability and instability for (complete) Kneser transversals. We first give a stability result for collections of $d+2(k-\lambda)$ points in $\mathbb{R}^d$ with $k-\lambda\geqslant 2$ and $\lambda =2,3$. We then present a description of Kneser transversals $L$ of collections of $d+2(k-\lambda)$ points in $\mathbb{R}^d$ with $k-\lambda\geqslant 2$ for $\lambda =2,3$. We show that either $L$ is a complete Kneser transversal or it contains $d-2(\lambda-1)$ points and the remaining $2(k-1)$ points of $X$ are matched in $k-1$ pairs in such a way that $L$ intersects the corresponding closed segments determined by them. The latter leads to new upper and lower bounds (in the case when $\lambda =2$ and $3$) for $m(k,d,\lambda)$ defined as the maximum positive integer $n$ such that every set of $n$ points (not necessarily in general position) in $\mathbb{R}^{d}$ admit a Kneser transversal. Finally, by using oriented matroid machinery, we present some computational results (closely related to the stability and unstability notions). We determine the existence of (complete) Kneser transversals for each of the $246$ different order types of configurations of $7$ points in $\mathbb{R}^3$.

In this paper, we investigate three problems concerning the toric ideal associated to a matroid. First, we list all matroids $\mathcal{M}$ such that its corresponding toric ideal $I_{\mathcal{M}}$ is a complete intersection. Second, we handle the problem of detecting minors of a matroid $\mathcal{M}$ from a minimal set of binomial generators of $I_{\mathcal{M}}$. In particular, given a minimal set of binomial generators of $I_{\mathcal{M}}$ we provide a necessary condition for $\mathcal{M}$ to have a minor isomorphic to $\mathcal{U}_{d,2d}$ for $d \geq 2$. This condition is proved to be sufficient for $d = 2$ (leading to a criterion for determining whether $\mathcal{M}$ is binary) and for $d = 3$. Finally, we characterize all matroids $\mathcal{M}$ such that $I_{\mathcal{M}}$ has a unique minimal set of binomial generators.

Given [Formula: see text], what is the number of numerical semigroups [Formula: see text] in [Formula: see text] of genus [Formula: see text]? After settling the case [Formula: see text] for all [Formula: see text], we show that attempting to extend the result to [Formula: see text] for all odd primes [Formula: see text] is linked, quite surprisingly, to the factorization of Fermat and Mersenne numbers.