7 |
7 |
8 %\title{Blob Homology} |
8 %\title{Blob Homology} |
9 \title{Sandbox} |
9 \title{Sandbox} |
10 \begin{document} |
10 \begin{document} |
11 |
11 |
12 \begin{align*} |
12 |
13 \begin{tikzpicture}[baseline] |
|
14 \node[draw] (c) at (0,0) [circle through = {(1,0)}] {$f$}; |
|
15 \node (d) at (c.east) [circle through = {(0.25,0)}] {}; |
|
16 \foreach \n in {1,2} { |
|
17 \node (p\n) at (intersection \n of c and d) {}; |
|
18 \fill (p\n) circle (2pt); |
|
19 } |
|
20 \begin{scope}[decoration={brace,amplitude=10,aspect=0.5}] |
|
21 \draw[decorate] (p2.east) -- node[right=2ex] {$a$} (p1.east); |
|
22 \end{scope} |
|
23 \end{tikzpicture} & = |
|
24 \begin{tikzpicture}[baseline] |
|
25 \node[draw] (c) at (0,0) [circle through = {(1,0)}] {}; |
|
26 \begin{scope} |
|
27 \path[clip] (c) circle (1); |
|
28 \node[draw,dashed] (d) at (c.east) [circle through = {(0.25,0)}] {}; |
|
29 \foreach \n in {1,2} { |
|
30 \node (p\n) at (intersection \n of c and d) {}; |
|
31 } |
|
32 \node[left] at (c) {$f$}; |
|
33 \path[clip] (d) circle (0.75); |
|
34 \foreach \y in {1,0.86,...,-1} { |
|
35 \draw[green!50!brown] (0,\y)--(1,\y); |
|
36 } |
|
37 \end{scope} |
|
38 \draw[->,blue] (1.5,-1) node[below] {$a \times I$} -- (0.75,0); |
|
39 \end{tikzpicture} \\ |
|
40 \begin{tikzpicture}[baseline] |
|
41 \node[draw] (c) at (0,0) [ellipse, minimum height=2cm,minimum width=2.5cm] {}; |
|
42 \draw[dashed] (c.north) -- (c.south); |
|
43 \node[right=6] at (c) {$g$}; |
|
44 \node[left=6] at (c) {$f$}; |
|
45 \end{tikzpicture} & = |
|
46 \begin{tikzpicture}[baseline] |
|
47 \node[draw] (c) at (0,0) [ellipse, minimum height=2cm,minimum width=2.5cm] {}; |
|
48 \node[right=9] at (c) {$g$}; |
|
49 \node[left=9] at (c) {$f$}; |
|
50 \draw[dashed] (c.north) to[out=-115,in=115] (c.south) to[out=65,in=-65] (c.north); |
|
51 \begin{scope} |
|
52 \path[clip] (c.north) to[out=-115,in=115] (c.south) to[out=65,in=-65] (c.north); |
|
53 \foreach \y in {1,0.86,...,-1} { |
|
54 \draw[green!50!brown] (-1,\y)--(1,\y); |
|
55 } |
|
56 \end{scope} |
|
57 \draw[->,blue] (.75,-1.25) node[below] {$a \times I$} -- (0,-0.25); |
|
58 \end{tikzpicture} \\ |
|
59 \begin{tikzpicture}[baseline] |
|
60 \node[draw] (c) at (0,0) [ellipse, minimum height=2cm,minimum width=2.5cm] {}; |
|
61 \draw[dashed] (c.north) -- (c.south); |
|
62 \node[right=18] at (c) {$g$}; |
|
63 \node[left=10] at (c) {$f$}; |
|
64 \fill (0,0.4) node (p1) {} circle (2pt); |
|
65 \fill (0,-0.4) node (p2) {} circle (2pt); |
|
66 \begin{scope}[decoration={brace,amplitude=5,aspect=0.5}] |
|
67 \draw[decorate] (p1.east) -- node[right=0.5ex] {\scriptsize $a$} (p2.east); |
|
68 \end{scope} |
|
69 \end{tikzpicture} & = |
|
70 \begin{tikzpicture}[baseline] |
|
71 \node[draw] (c) at (0,0) [ellipse, minimum height=2cm,minimum width=2.5cm] {}; |
|
72 \node[draw,dashed] (d) at (0,0) [circle, minimum height=1cm,minimum width=1cm] {}; |
|
73 \draw[dashed] (c.north) -- (d.north) (d.south) -- (c.south); |
|
74 \node[right=18] at (c) {$g$}; |
|
75 \node[left=18] at (c) {$f$}; |
|
76 \clip (0,0) circle (0.5cm); |
|
77 \foreach \y in {1,0.86,...,-1} { |
|
78 \draw[green!50!brown] (-1,\y)--(1,\y); |
|
79 } |
|
80 \end{tikzpicture} \\ |
|
81 \end{align*} |
|
82 |
13 |
83 \begin{tikzpicture} |
14 \begin{tikzpicture} |
84 \node[circle,fill=black,inner sep=1pt] (A) at (1.73,0) {}; |
15 \node[circle,fill=black,inner sep=1pt] (A) at (1.73,0) {}; |
85 \node[circle,fill=black,inner sep=1pt] (B) at (-1.73,0) {}; |
16 \node[circle,fill=black,inner sep=1pt] (B) at (-1.73,0) {}; |
86 \draw[dashed] (A) -- (B); |
17 \draw[dashed] (A) -- (B); |