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Knots such as the 3_1 knot and the 4_1 knot are often referred to as the trefoil and figure-eight knots respectively. There are more obscure names for some of the later ones in the knot tables, for example the 6_1 knot is also know as the stevedore knot. These names are not listed on the online knot database at http://www.indiana.edu/~knotinfo/ are they listed anywhere else?

Is there an (online) database of knots that includes their (text) names.

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Yup: katlas.org/wiki/Main_Page Also, if you want a non on-line source, the Ashley Book of Knots has loads of named knots, although many of them are friction knots and splices so the density of mathematical knots is relatively low. –  Ryan Budney Sep 24 '10 at 23:53
    
Also, the Indiana "knotinfo" webpage produces links to Dror Bar-Natan's webpage if you request it to. –  Ryan Budney Sep 25 '10 at 0:08
    
Sorry, I was looking for a large table listing the knots and their alternate names similar to the information / format provided by knotinfo. I see katlas.org has alternate names listed on the individual pages of knots but the "take home database" doesn't seem to include this. –  Mark Bell Sep 25 '10 at 0:13
    
Ah, that's weird. Scott Morrison might be able to help as I believe he play(s/ed) a role in setting up the katlas page. –  Ryan Budney Sep 25 '10 at 0:36

1 Answer 1

As the comments suggest, the tables enumerated knots are much larger than any table of "named" knots. So it might be easier to start with the "named" knots and identify them in the table. The snappy software (http://www.math.uic.edu/t3m/SnapPy/index.html) allows you to do this quite easily for hyperbolic knots. Here is a general outline of how to do that. (snappy: "text" are instructions to give snappy comments have a bullet in front of them.)

snappy: M = Manifold()

  • If you have installed plink a window pops up where you can draw your knot

snappy: M.solution_type()

  • This should be 'all tetrahedra positively oriented' if you have something hyperbolic

snappy: CK = CensusKnots()

  • This loads the list of knots known to decompose into 8 or fewer tetrahedra

snappy: CK.identify(M)

  • This will return the name of the manifold in the census you are looking at.

You can also get something similar to work if you want to look at the AlternatingKnotExteriors or NonalternatingKnotExteriors. Unfortunately, the identify function won't work on these list, but you can you can still get something to work.

snappy: M = Manifold()

snappy: M.solution_type()

  • Again begin the same way and check that you have a hyperbolic knot that admits a diagram with fewer than 16 crossings.

snappy: volM = M.volume()

snappy: for k in AlternatingKnotExteriors():

snappy: if k.volume() < volM + .002 and k.volume() > volM - .002:

snappy: print k

snappy: print k.is_isometric_to(M)

  • This will give you a good list of what your knot could be. You could also do the same thing for the NonalternatingKnotExteriors(). Finally, the .002 is there to account for any error that might arise computationally. In principle, you could probably choose a number much smaller.
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Thanks for the suggestion. Just to let you know you can actually skip the volume filtering, Dr. Weeks built it directly into the snappea kernel! Line 90 of isometry.c defines "CRUDE_VOLUME_EPSILON" to be 0.01. Later, around line 140, when testing if two manifolds are isometric if they differ by more than this amount the entire calculation is aborted and the function returns "not isometric". –  Mark Bell Feb 28 '13 at 22:52

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