5
$\begingroup$

2-bridge knots (aka rational knots) $K(p,q)$ are described by a rational number $\frac{p}{q}$ or likewise its continued fraction expansion $\left[a_1,a_2,\ldots,a_k\right]$.

Has somebody worked out a list to identify the 2-bridge knots in the Rolfsen's table or the Callahan-Hildebrand-Weeks census or some other knot census?

This would be helpful to read the invariants (Alexander polynomial, hyperbolic volume etc) of $K(p,q)$ for various $\frac{p}{q}$ from one of these tables.

$\endgroup$
12
$\begingroup$

Cha and Livingston's KnotInfo includes the ability to list the bridge index of knots with at most 11 crossings, and so covers all of Rolfsen's table. The 2-bridge knots are those with bridge index 2. Additionally, their table includes the other invariants that you mention (Alexander polynomial, hyperbolic volume) as well as many more.

In the description of bridge index they say that ''There is no general method or algorithm for computing the bridge index of an arbitrary knot''. However, the case of knots with at most 11 crossings was done by Musick in ''Minimal Bridge Projections for 11-corssing prime knots''. In this paper he provides an explicit Dowker-Thistlethwaite Code for a diagram of minimal bridge index for each 11 crossing knots in Section 3. From which one could compute $p / q$ from directly.

$\endgroup$
  • $\begingroup$ Thank you. <a href="indiana.edu/~knotinfo/… list</a> is exactly what I wanted. $\endgroup$ – ThiKu Mar 13 '15 at 8:05
  • 1
    $\begingroup$ In fact, it also includes the Two-bridge notation invariant, which is very convenient. $\endgroup$ – Marco Golla Mar 13 '15 at 8:19
6
$\begingroup$

Mark Bell has a wonderful answer for the knot tables. I thought I would provide an answer for the census data. Using SnapPy, the current version of the OrientableCuspedCensus, which is a combination of the Callahan-Hildebrand-Weeks census, Morwen Thistlethwaite's 8-tetrahedral census and Ben Burton's 9 tetrahedral census, appears to have 77 2-bridge knots according the following computation.

 CC = OrientableCuspedCensus(num_cusps=1)
 knotcount = 0
 for C in CC:
    if C.is_two_bridge():
            print C.identify(), C.is_two_bridge(), C.volume()
            knotcount = knotcount+1
 print knotcount

This function runs in the SnapPy application. Note: printing C.is_two_bridge() identifies the two bridge knot. Note that copying the code and then typing '%paste' on the command line is the easiest way to run the code yourself.

$\endgroup$
1
$\begingroup$

Rolfsen's diagrams of 2-bridge knots are easy to see (from Conway's tangle notation(. I had no problem listing them first in my 1974 table of 10-crossing knots, although those diagrams were Tait's. Indeed, it's remarkable that Schubert missed three of them through 9 crossings in his seminal paper introducing Schubert normal form (a real monstrosity from the diagramming standpoint).

$\endgroup$
  • 2
    $\begingroup$ Could you please give more information so that readers could find your 1974 table? $\endgroup$ – Zach Teitler Jul 6 '17 at 2:54

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.