Burau Representation

Author

Eric W. Weisstein
January 23, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/KnotTheory/BurauRepresentation.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/BurauRepresentation.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Code

Burrau[n_Integer,i_Integer]:=Module[{m=IdentityMatrix[n-1]},
m[[i,i]]=-t;
If[i>1,m[[i-1,i]]=-t];
If[i<n-1,m[[i+1,i]]=-1];
m
]/;0<i<n

Examples

Burrau[5,1]//TraditionalForm

Burrau[5,2]//TraditionalForm

Burrau[5,4]//TraditionalForm

Knots

Solomin's Seal

#.#.#.#.#&@Burrau[5,1]//TraditionalForm

Det[IdentityMatrix[4]-#.#.#.#.#&@Burrau[5,1]]

Trefoil Knot

Burrau[3,1]//TraditionalForm

#.#.#&@Burrau[3,1]//TraditionalForm

IdentityMatrix[2]-Dot[#,#,#]&@Burrau[3,1]//Expand

((Det[IdentityMatrix[2]-Dot[#,#,#]&@Burrau[3,1]])/(1+t+(t^2)))