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Has anybody done any work on making a classification of low-complexity tangles, analogous to the work for knots and links? I expect most of the small ones to be rational, and those that aren't rational to be decomposable in the way the square knot is decomposable. But there are more, and I'd like to see a list of the small ones; in particular, the non-mathematician usage of "knot" is often referring to something that could be formalized like this.

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  • $\begingroup$ I can't find the sources at the moment, but people have made tables of spatial graphs with low crossing number. These correspond, in some sense, to tangles in that the exterior of an edge of the graph is a tangle. $\endgroup$ – Scott Taylor Aug 19 '14 at 15:37
  • $\begingroup$ Scott, I'd be interested in these tables if you can find them. $\endgroup$ – Dylan Thurston Aug 19 '14 at 15:49
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As Sam Nead points out, there is a relationship between spatial graphs and tangles obtained by drilling out one edge of a spatial graph having two vertices or drilling out the vertex of a spatial graph having a single vertex. There have been a few people working on building tables of spatial graphs. Here are a few references:

Moriuchi "An enumeration of theta curves with up to 7 crossings" (MR2507922)
Ishii, et al "A table of genus 2 handlebody knots up to 6 crossings" (MR2890461)
Lee and Lee "Inequivalent handlebody-knots with homeomorphic complements"(MR2928904)

A handlebody knot is the regular neighborhood of a spatial graph and equivalence of handlebody knots corresponds to neighborhood equivalence of spatial graphs.

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  • $\begingroup$ Thanks! More directly relevant is a masters thesis by Yamano mentioned in the Moriuchi paper. That thesis is in Japanese, but Google finds this paper: $\endgroup$ – Dylan Thurston Aug 20 '14 at 1:49
  • $\begingroup$ Kanenobu, Saito, Satoh, "Tangles with up to Seven Crossings", Interdisciplinary Information Sciences, Vol. 9, No. 1, 127-140. That is a bizarre place to publish, and MR hasn't indexed it. But it's also really not going very far; their table only has 23 connected tangles. They claim Yamano has done up to 9 crossings. $\endgroup$ – Dylan Thurston Aug 20 '14 at 1:59
  • $\begingroup$ They have a somewhat odd notion of equivalence. For instance, the sheet bend is $6_4$ in their table, and won't look familiar to anyone used to knots. $\endgroup$ – Dylan Thurston Aug 20 '14 at 2:00
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Nice question! I don't know of such a census. Probably the first course of action is to email Damien Heard, the author of the program "Orb". Here is the page for Orb.

http://www.ms.unimelb.edu.au/~snap/orb.html

The documentation, a 13 page pdf file in the source code archive, is very helpful. Basically, you want to draw a theta-graph, drill one of the edges, and label each of the other edges with a "2". These graphs are in one-to-one correspondence with tangles (well, up to a Dehn twist about the equator of the tangle).

Orb can find canonical triangulations for these and can, it appears, use them to check for isometries. This, in addition to its ability to compute volumes and length spectra, means it can deal with the filtering side of census building. Orb can also compute two-fold branched covers, which will be handy.

I guess some version of DT codes will deal with the generation side of census building.

Needless to say, this sounds like a non-trivial amount of work...

Postscript: I'd be very interested to see how the two-fold branched covers of tangles arrange themselves in the SnapPea cusped census. Is there a way to directly "see" which elements of the cusped census have tangle quotients?

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  • $\begingroup$ Thanks, Sam! It looks like a nice program. As you say, it's only part of what I'm looking for. $\endgroup$ – Dylan Thurston Aug 19 '14 at 15:50
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Have you seen the paper "Matrix Integrals and the Generation and Counting of Virtual Tangles and Links"?

  • P. Zinn-Justin and J.-B. Zuber, Matrix Integrals and the Generation and Counting of Virtual Tangles and Links, Journal of Knot Theory and Its Ramifications, 13:03, May 2004. arXiv

Associated with that paper there is also a database on Zinn-Justin's website, which includes diagrams and information about prime virtual alternating links (resp. tangles) with up to 8 (resp. 6) crossings.

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