add new image; fix labelling in old image

posts/chebyshev/2/index.qmd

@@ -1,12 +1,12 @@
 ---
 title: "Generating Polynomials, Part 2: Ghostly Chains"
 description: |
-  "Do polygons without distance still know about planar geometry?"
+  "What do polygons without distance still know about planar geometry?"
 format:
   html:
     html-math-method: katex
 date: "2021-08-19"
-date-modified: "2025-06-20"
+date-modified: "2025-06-24"
 categories:
   - geometry
   - algebra
@@ -290,7 +290,6 @@ Powerful Chains
 
 The various $P_n$ are in fact the adjacency matrices of a path on *n* nodes.
 
-<!-- TODO: bad labelling in figure -->
 ![
   Example path graphs of orders 2, 3, and 4
 ](./path_graphs.png)
@@ -506,7 +505,10 @@ In some sense, they are the opposite of cycle graphs, since by definition they c
 Paths are degenerate trees, but we can make them slightly more interesting by instead adding
   exactly one node and edge to (the middle of) a path.
 
-<!-- TODO: new image for T_{1,1} through T_{2,2} -->
+![
+  Nondegenerate tree graphs based on 3-, 4-, 5-, and 6-paths
+](./tree_graphs.png)
+
 $$
 \begin{align*}
   T_{1,1} := \begin{matrix}[
@@ -774,12 +776,13 @@ $$
 Searching the OEIS for the coefficients of $t_{2,4}$ returns sequence
   [A228786](http://oeis.org/A228786), which informs that it is the minimal polynomial
   of $2\sin( \pi/15 )$.
-This sequence also informs that the other factor is the minimal polynomial of $2\sin( \pi / 9 )$:
+This sequence also informs that the unknown factor in the other polynomial is
+  the minimal polynomial of $2\sin( \pi / 9 )$:
 
-In fact, both of these polynomials show up in the factorization
-  of Chebyshev polynomials of the *first* kind.
-Perhaps this is not surprising, given how $2\cos(\pi / n)$ also appear in the spectra of graphs.
-However, it is immensely interesting to see them pop out from the addition of a single node.
+In fact, both of these polynomials show up in factorizations of Chebyshev polynomials
+  of the *first* kind (specifically, $2T_15(z / 2)$ and $2T_9(z / 2)$).
+Perhaps this is not surprising since we were already working with those of the second kind.
+However, it is interesting to see them appear from the addition of a single node.
 
 
 Closing
@@ -796,5 +799,5 @@ Regardless, they still related to Chebyshev polynomials, albeit through their fa
 
 In fact, I was initially prompted to look into them due to a remarkable correspondence between
   certain trees and Platonic solids.
-I have reorganized these thoughts, since from the perspective of this article, the relationship
+I have since reorganized these thoughts, as from the perspective of this article, the relationship
   is tangential at best.