Today. EIWL 5-8, a riddle on spheres, more on Catalan, and maybe, Etienne’s project.

Topics (in no particular order). Whatever you may suggest; whatever comes to my mind; the Fibonacci numbers; the Catalan numbers; the Jones polynomial; a more efficient Jones algorithm; a riddle on spheres; Khovanov homology; Γ-calculus; the Hopf fibration; Hilbert’s 13th problem; non-commutative Gaussian elimination; free Lie algebras; the Baker-Campbell-Hausdorff formula; wacky numbers; an order 4 torus; the Schwarz Lantern; knot colourings; the Temperley-Lieb pairing; the dodecahedral link; sound experiments; barycentric subdivisions; a Peano curve; braid closures and Vogel’s algorithm; the insolubility of the quintic; phase portraits; the Mandelbrot set; shadows of the Cantor aerogel; quilt plots; some image transformations; De Bruijn graphs; the Riemann series theorem; finite type invariants and the Willerton fish.

Pensieve header: October 2: More on the Catalan numbers.

The Catalan Numbers

Project Idea. Make an easily extendible, n-dependent “Catalan objects poster”, as explained on the blackboard.

ts[n_Integer]:=ts[Range[0,n+1]];
ts[{_,_}]={ds[]};
ts[vs_List] := Module[{l,r,k,t1,t2,tds},
Union@@Table[
l=ts[Prepend[vs[[k;;]],vs[[1]]]];
r=ts[vs[[2;;k]]];
Flatten[Table[
tds=Join[t1,t2];
If[k>3, AppendTo[tds, d[vs[[2]],vs[[k]]]]];
If[k<Length[vs], AppendTo[tds, d[vs[[1]],vs[[k]]]]];
tds,
{t1,l}, {t2,r}
]],
{k,3,Length[vs]}
]
]

ts[4]