Why such an ugly Braid Representative?

From Knot Atlas
Jump to: navigation, search

See an email exchange on L4a1's "ugly" braid representative:

An email inquiry by Richard Kubelka

Date: Wed, 9 Mar 2011 12:48:47 -0800
From: Richard Kubelka <...>
To: drorbn@math.toronto.edu
Subject: Braid representative for L4a1 in the Knot Atlas

Professor Bar-Natan:
Re the braid representative for L4a1 in the Knot Atlas (see
http://katlas.math.toronto.edu/wiki/L4a1 ).
Is there any particular reason the representative [1 -2 3 -2 -1 -2 -3
-2] is given rather than the simpler [1 1 1 1] (or [-1 -1 -1 -1])?
The other similar links L2a1, L6a3, L8a14, L10a118 in the table are
shown with the simpler braid representatives [1 1], [1 1 1 1 1 1], [1 1
1 1 1 1 1 1], and [1 1 1 1 1 1 1 1 1 1] respectively.
I realize that as oriented links, [1 -2 3 -2 -1 -2 -3 -2]  and [1 1 1 1]
are not the same, but since the table
http://katlas.math.toronto.edu/wiki/The_Thistlethwaite_Link_Table is not
a table of oriented links, why not choose the simpler braid
representative?  Am I missing something?

Rick Kubelka

Richard P. Kubelka, Ph.D.
Professor and Graduate Coordinator
Department of Mathematics
San Jose State University
San Jose, CA  95192-0103

(...) ...-.... voice
(...) ...-.... fax

Response by Dror Bar-Natan

Date: Fri, 25 Mar 2011 13:22:56 -0400 (EDT)
From: Dror Bar-Natan <drorbn@math.toronto.edu>
To: Richard Kubelka <...>
Cc: Scott Morrison <...>
Subject: Re: Braid representative for L4a1 in the Knot Atlas

Dear Rick Kubelka,

Sorry for the time it took me to respond.

There are two answers to your question:

1. The Thistlethwaite table is indeed a table of unoriented links, but for each
link appearing in the knot atlas some orientation is implicitly chosen in an
arbitrary manner, and then all invariants are computed with that given
orientation. Many invariants are actually invariants of oriented links. You can
always see which orientation was chosen for any given link by looking at the
"Morse Link Representative".

2. Actually, with the exception of the knots in the Rolfsen table, braid
representatives were computed by a straight-forward implementation of Vogel's
algorithm, without following it by any simplifications. So many braid
representatives are sub-optimal.

Would you allow me to post your question and my answer at
http://katlas.org/w/index.php?title=Notes_on_L4a1's_Link_Presentations? There
may be some wider interest in our discussion.