A self-dual map $G$ is said to be \emph{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 \emph{involutive labelings}. The latter lead us to obtain necessary conditions for a map to be \emph{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 \emph{ antipodally symmetric} maps. It turns out that the latter is a very helpful tool to study questions concerning the \emph{symmetry} as well as the \emph{amphicheirality} of \emph{links}.

We give a closed formula to evaluate exterior webs (also called MOY webs) and the associated Reshetikhin-Turaev link polynomials.

A subset $S$ of $\mathbb R^d$ has the Borsuk property if it can be decomposed into at most $d+1$ parts of diameter smaller than $S$. This is an important geometric property, inspired by a conjecture of Borsuk from the 1930s, which has attracted considerable attention over the years. In this paper, we define and investigate the Borsuk property for matroids, providing a purely combinatorial approach to the Borsuk property for matroid polytopes, a well-studied family of $(0,1)$-polytopes associated with matroids. We show that a sufficient condition for a matroid -- and thus its matroid polytope -- to have the Borsuk property is that the matroid or its dual has two disjoint bases. However, we show that this condition is not necessary by exhibiting infinite families of matroids having the Borsuk property and yet being such that every two bases intersect and every two cobases intersect. Kneser graphs, which form an important object from topological combinatorics, play a crucial role in most proofs.