This talk describes the combinatorics and geometry related to a family of subvarieties of the flag variety called {\em Springer fibers}. We work in type $A_{n-1}$, in which the flag variety is the collection of nested vector subspaces $V_1 \subseteq V_2 \subseteq \cdots V_{n-1} \subseteq \mathbb{C}^n$ of a fixed complex vector space. The Springer fiber of a linear operator $X$ consists of those flags that are fixed by $X$ in the sense that $XV_i \subseteq V_i$ for all $i$. Springer fibers are the ``next simplest” subvarieties of the flag variety after Schubert varieties; yet, unlike Schubert varieties, very little is known about their general topology other than a cell decomposition.

We give new work about closures of these cells for certain Springer fibers. The combinatorial objects that most naturally characterize closure relations in our cases come from the intersection of knot theory and combinatorial representation theory. Khovanov first observed that when $X$ has two Jordan blocks of the same size, the cohomology ring of the associated Springer fiber encodes important aspects of the Temperley-Lieb algebra. More generally, the cells of Springer fibers with $k$ equal-sized Jordan blocks are related to webs, the morphisms in a diagrammatic category encoding quantum representations of $\mathfrak{sl}_k$. The relationship between webs and Springer fibers is very well-understood when $X$ has two equal-sized Jordan blocks, somewhat understood when $X$ has three, and mysterious when $X$ has more blocks. Time permitting, we will finish with conjectures that might extend to these mysterious cases.