Pensieve header: October 23: Some further Hochschild Homology.

Today. Some further Hochschild homology, then whatever you may suggest, then maybe EIWL 9-12, then, if we’re kidding ourselves, Patterns.

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; the Towers of Hanoi; Hochschild homology of (some) coalgebras; convolutions and image improvements.

An Image Manipulation Challenge

The image at http://drorbn.net/bbs/show?shot=17-1750-171016-111042.jpg is pathetic. Can you improve it? Whatever you do, should also work well with all other images at http://drorbn.net/bbs/show.php?prefix=17-1750.

Some Hochschild Homology

First see the image at http://drorbn.net/AcademicPensieve/Classes/17-1750-ShamelessMathematica/index.html?im=171023-HomologyBBS.png.

(*(Subscript[d, n_,k_])[ℰ_] := ℰ /. {(Subscript[x, i_])/;i<k:>(Subscript[x, i]), (Subscript[x, i_])/;i==k:>(Subscript[x, k])+(Subscript[x, k+1]), (Subscript[x, i_])/;i>k:>(Subscript[x, i+1])};*)
(Subscript[d, n_,k_])[ℰ_] := ℰ /. (Subscript[x, i_]):>Which[i<k, (Subscript[x, i]),i==k, (Subscript[x, k])+(Subscript[x, k+1]), i>k, (Subscript[x, i+1])];
(Subscript[d, n_])[ℰ_] := Expand@Sum[(((-1))^k)(Subscript[d, n,k])[ℰ], {k,0,n+1}];
(Subscript[C, 0,p_]):=If[p==0,{1}, {}];
(Subscript[C, n_,p_]):=Union@@Table[(Subscript[x, n]^k)(Subscript[C, n-1,p-k]), {k,0,p}];

?Switch

o[x_] := Switch[x, 2,3,4|5,6,_,7];
o /@ Range[10]

(Subscript[C, 3,4])

(Subscript[d, 3])/@(Subscript[C, 3,4])

(Subscript[d, 4])/@ (Subscript[d, 3])/@(Subscript[C, 3,4])

(Subscript[M, n_,p_]):=Table[Coefficient[(Subscript[d, n])[a],b],{b,(Subscript[C, n+1,p])},{a,(Subscript[C, n,p])}];
(Subscript[β, n_,p_]):=Length[NullSpace[(Subscript[M, n,p])]]-MatrixRank[(Subscript[M, n-1,p])];

Table[(Subscript[β, n,p]),{n,1,5},{p,1,5}] // MatrixForm // Timing