\[out=up, in=up, looseness=2.5\] (2,0) ; at (2.0,0) [$\pv V$]{}; at (0,0) [$V^*$]{};

&

(0,0) to \[out=up, in=up, looseness=2.5\] (2,0) ; at (2.05,0) [$\pv V$]{}; at (0,0) [$V^*$]{};

\
fv f(v) &  1\_V & vff(v) &   1\_V

To define the second and fourth map, we have identified $V\otimes V^* \cong V^*\otimes V \cong\End(V)$. It may be verified that these maps fulfill the following *snake equations*:

\[eq:snake\]

(0,0) to (0,1) to \[out=up, in=up, looseness=2\] (1,1) to \[out=down, in=down, looseness=2\] (2,1) to (2,2);

   =   

(0,0) to (0,2);

   =    

(0,0) to (0,1) to \[out=up, in=up, looseness=2\] (1,1) to \[out=down, in=down, looseness=2\] (2,1) to (2,2);

&

(0,0) to (0,1) to \[out=up, in=up, looseness=2\] (1,1) to \[out=down, in=down, looseness=2\] (2,1) to (2,2);

    =   

(0,0) to (0,2);

   =    

(0,0) to (0,1) to \[out=up, in=up, looseness=2\] (1,1) to \[out=down, in=down, looseness=2\] (2,1) to (2,2);

Together with the swap map $\sigma_{V,W}:\ignore{V\otimes W\to W\otimes V$,~$}v\otimes w\mapsto w\otimes v$, depicted as a crossing of wires, this leads to an extremely flexible topological calculus, allowing us to untangle arbitrary diagrams and straighten out any twists:

&

(0,0) to \[out=up, in=right\] (-1.25, 2.25) to \[out=left, in=left, looseness=1\] (-1.25,0.75) to \[out=right, in=down\] (0,3);

   =   

(0,0) to (0,3);

   =   

(0,0) to \[out=up, in=right\] (-1.25, 2.25) 