Dror Bar-Natan: Papers: WKO:

The Kishino Braid

Pensieve Header: We verify that the "Kishino Braid" is non-trivial, yet its Alexander invariant is trivial.

Dear Jana, Karene, Lucy, Peter and Zsuzsi,

In the last Knot at Lunch meeting we mentioned the fact that probably some
virtual braids cannot be seen by their action on Aut(F_n). Well, I've just
verified that a braid that Karene and I were talking about, the "Kishino
Braid" is non-trivial yet its projection into both w-braids and
mirror-w-braids is trivial. So both actions of that braid on the free
group are trivial.

See a mathematica notebook (also in pdf) at
http://katlas.math.toronto.edu/drorbn/AcademicPensieve/2008-12/ (under
"mathematica notebooks", not under "notebook pages"!).

The fact that the Kishino braid is both w-trivial and mirror-w-trivial is
visibly obvious. So the only problem is to check that it is a non-trivial
v-braid, and this I do by computing its Jones polynomial. As sanity checks
I use Jana's thesis to compute its Alexander and the Alexander of the
mirror (both are trivial, as expected). Another sanity check is to carry
out the same computations for a trivial braid, and they all work out as
they should.

Best,

Dror.

<< KnotTheory`

B = PD[
X[18,2,19,1], X[27,3,28,2], X[19,3,20,4], X[28,4,29,5],
X[5,20,6,21], X[6,29,7,30], X[7,22,8,21], X[8,31,9,30],
X[31,10,32,9], X[22,11,23,10], X[32,11,33,12], X[23,12,24,13],
X[13,33,14,34], X[14,24,15,25], X[15,35,16,34], X[16,26,17,25]
];

The Jones is non-trivial!

Jones[B][q] // Short

Yet the Alexander, also of the Mirror, is trivial:

Import["http://www.math.toronto.edu/~drorbn/Talks/Sandbjerg-0810/pA.m"]

CircuitDiagram[B_PD] := CircuitDiagram @@ B /. {
X[i_,j_,k_,l_] /; j>l :> Xp[i,j,k,l],
X[i_,j_,k_,l_] /; j<l :> Xm[i,j,k,l]
}

B1=CircuitDiagram[B]

pA[B1]

B2=CircuitDiagram[Mirror[B]]

pA[B2]

pA[B1]==pA[B2]

Last[pA[B1]] // Factor

Half of B

Jones[Take[B, 8]][q]

pA[CircuitDiagram[Take[B,8]]] // Last // Factor

pA[CircuitDiagram[Mirror[Take[B,8]]]] // Last // Factor

Sanity check on a trivial braid:

TB=PD[X[1,6,2,7], X[2,9,3,10], X[3,11,4,10], X[4,8,5,7]]

{Jones[TB][q], Jones[Mirror[TB]][q]}

{TB1=CircuitDiagram[TB], TB2=CircuitDiagram[Mirror[TB]]}

pA[TB1]

pA[TB2]

Last[pA[TB1]] // Factor

Last[pA[TB2]] // Factor