Burau's Theorem

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An interesting property of the Alexander polynomial related to cables is Burau's theorem which says the following: If we take the $n$-th cable of a knot from the Knot Atlas and insert in it a braid with $1/n$ of a full twist, the Alexander polynomial of the result with respect to $t$ is the same as the Alexander polynomial of the original knot with respect to $t^{n}$. This can only be seen if we take into consideration the writhe $w$ of the knot and add the appropriate number of twists ($nw(1/n)=w$ full twists) in the direction opposite to the sign of the writhe. We can test the theorem using the program CableComponent and performing the above operation on knot K11n152, for example.

(For In[1] see Setup)

In[2]:=

Import["http://katlas.org/wiki/CableComponent.m&action=raw"];

In[3]:=	(cc = CableComponent[BR[6, {1, 2, 3, 4, 5}], K = Mirror[Knot[11, NonAlternating, 52]]]) // DrawMorseLink
	File:Burau's Theorem Out 3.gif
Out[3]=	-Graphics-

It is not immediately clear from the diagram that this is the expected result but we can see that the Alexander polynomial relation holds:

In[4]:=	Alexander[#][t] & /@ {cc, K}
Out[4]=	-18 6 14 6 12 18 {-17 + t - --- + -- + 14 t - 6 t + t , 12 6 t t -3 6 14 2 3 -17 + t - -- + -- + 14 t - 6 t + t } 2 t t