📂Topological Data Analysis

Definition of Simplicial Complexes

Definition

Difficult Definition ¹

$$\Delta^{k} \in K$$

A complex is called a Simplicial Complex if a finite set of simplices $K$ satisfies the following two conditions:

• (i): If $\sigma \in K$ and $\tau$ is a face of $\sigma$, then $\tau \in K$. $$\sigma \in K \land \tau \le \sigma \implies \tau \in K$$
• (ii): If $\sigma_{1}, \sigma_{2} \in K$, then $\sigma_{1} \cap \sigma_{2}$ is either an empty set or a face of both $\sigma_{1}$ and $\sigma_{2}$. $$\sigma_{1} , \sigma_{2} \in K \implies \left( \sigma_{1} \cap \sigma_{2} = \empty \right) \lor \left( \sigma_{1} \cap \sigma_{2} \le \sigma_{1} \land \sigma_{1} \cap \sigma_{2} \le \sigma_{2} \right)$$

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• $\land$ is logically the symbol for ‘and’, an and operation.
• $\lor$ is logically the symbol for ‘or’, an or operation.
• A face of simplex $x$ is a simplex created by removing one point from $x$.
• For simplices $\tau$, $\sigma$, $\tau \le \sigma$ means that $\tau$ is a face of $\sigma$.

Simple Definition

A Simplicial Complex is a collection of simplices, where every connecting part is a simplex itself.

Explanation

Simplices are meaningful and useful on their own, but by forming a Simplicial Complex, one can obtain an approximation of almost any abstract object with geometric characteristics. ² For example, the following is the Triangulation of a dolphin shape, which is a simplicial complex made up of maximum $2$-simplices (triangles).

The simple definition implies vaguely connecting sets, a notion often breezily introduced and passed over in many documents and lectures. This approach favors demonstrating practical and application aspects of the simplicial complex over a rigorous definition, as showing a figure tends to be more comprehensible and explanatory.

Naturally, for solo study with a book, one must precisely understand the challenging definitions. A simplicial complex $K$ is originally a family of sets of simplices $\Delta^{k}$, each being the convex hull of $k$ affinely independent points, thus allowing for considerations of intersections like $\sigma_{1} \cap \sigma_{2}$.

Polygon

By definition, Polygons look like simplicial complexes but are not since they include shapes like rectangles.

See Also

A simplicial complex is defined as such if a set $K$ satisfies all the given conditions, without specific restrictions on its exact appearance. Depending on how one defines simplices, countless complexes can be imagined, and even with the same points (data), the practical characteristics can greatly vary among simplicial complexes.