Why such an ugly Braid Representative?
From Knot Atlas
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: email@example.com 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? Thanks, 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 www.math.sjsu.edu/~kubelka
Response by Dror Bar-Natan
Date: Fri, 25 Mar 2011 13:22:56 -0400 (EDT) From: Dror Bar-Natan <firstname.lastname@example.org> 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. Best, Dror.