Burau's Theorem

KnotTheory`/Navigation  The Mathematica Package KnotTheory` Acknowledgement Setup Naming and Enumeration Presentations Planar Diagrams Some further details Gauss Codes DT (Dowker-Thistlethwaite) Codes Braid Representatives MorseLink Presentations Arc Presentations Conway Notation Graphical Input Graphical Output Drawing Planar Diagrams How does it work? Drawing MorseLink Presentations Drawing Braids Structure and Operations Invariants Invariants from Braid Theory Three Dimensional Invariants The Alexander-Conway Polynomial The Multivariable Alexander Polynomial The Determinant and the Signature The Jones Polynomial How is the Jones polynomial computed? The Coloured Jones Polynomials The A2 Invariant Quantum knot invariants The HOMFLY-PT Polynomial The Kauffman Polynomial Finite Type (Vassiliev) Invariants Khovanov Homology Heegaard Floer Knot Homology R-Matrix Invariants Extras Included with KnotTheory` Drawing with TubePlot Standalone_TubePlot Using the LinKnot package WikiLink QuantumGroups` Lightly Documented Features A Sample KnotTheory` Session Further Usage Example Prime Links with a Non-Prime Component "Rubberband" Brunnian Links Identifying Knots within a List Cabling Burau's Theorem Further Knot Theory Software KnotPlot Knotscape Xknots Extending/Modifying KnotTheory` Bugs How to Edit this Manual...

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/w/index.php?title=CableComponent.m&action=raw"];

In[3]:=	(cc = CableComponent[BR[6, {1, 2, 3, 4, 5}], K = Mirror[Knot[11, NonAlternating, 52]]]) // DrawMorseLink
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