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Abstract We study the open, closed, and non-degenerate embedding dimensions of neural codes, which are the smallest respective dimensions in which one can find a realization of a code consisting of convex sets that are open, closed, or non-degenerate in a sense defined by Cruz, Giusti, Itskov, and Kronholm. For a given code$$\mathcal {C}$$ $C$we define the embedding dimension vector to be the triple (a, b, c) consisting of these embedding dimensions. Existing results guarantee that$$\max {\{a,b\}}\le c$$ $max\{a,b\}\le c$, and we show that when any of these dimensions is at least 2 this is the only restriction on such vectors. Specifically, for every triple (a, b, c) with$$2\le \min {\{a,b\}}$$ $2\le min\{a,b\}$and$$\max {\{a,b\}}\le c\le \infty $$ $max\{a,b\}\le c\le \infty$we construct a code$$\mathcal {C}_{(a,b,c)}$$ $C_{(a,b,c)}$whose embedding dimension vector is exactly (a, b, c) (where an embedding dimension is$$\infty $$ $\infty$if there is no realization of the corresponding type). Our constructions combine two existing tools in the convex neural codes literature: sunflowers of convex open sets, and rigid structures, the latter of which was recently defined in work of Chan, Johnston, Lent, Ruys de Perez, and Shiu. Our constructions provide the first examples of codes whose closed embedding dimension is larger than their open embedding dimension, but still finite. 

https://arxiv.org/abs/2309.11237 

https://arxiv.org/abs/2309.14862 

The augmented Bergman complex of a closure operator on a finite set interpolates between the order complex of proper flats and the independence complex of the operator. In 2020, Braden, Huh, Matherne, Proudfoot, and Wang showed that augmented Bergman complexes of matroids are always gallery-connected, and recently Bullock, Kelley, Reiner, Ren, Shemy, Shen, Sun, Tao, and Zhang strengthened gallery-connected to shellable by providing two classes of shelling orders: flag-to-basis shellings and basis-to-flag shellings.  We show that augmented Bergman complexes of matroids are vertex decomposable, a stronger property than shellable. We also prove that the augmented Bergman complex of any closure operator is shellable if and only if the order complex of its lattice of flats (that is, its non-augmented Bergman complex) is shellable. As a consequence, an augmented Bergman complex is shellable if and only if it admits a flag-to-basis shelling. Perhaps surprisingly, the same does not hold for basis-to-flag shellings: we describe a closure operator whose augmented Bergman complex is shellable, but has no shelling order with bases appearing first.  

We prove algebraic and combinatorial characterizations of the class of inductively pierced codes, resolving a conjecture of Gross, Obatake, and Youngs. Starting from an algebraic invariant of a code called its canonical form, we explain how to compute a piercing order in polynomial time, if one exists. Given a piercing order of a code, we explain how to construct a realization of the code using a well-formed collection of open balls, and classify the minimal dimension in which such a realization exists. 

https://arxiv.org/abs/2205.04776 

https://arxiv.org/abs/2203.12063