Update 03/2008: links with the topology of a torus (genus 1) now have an alternate picture as a diagram drawn on a torus viewed as the doubly periodic plane. There are still some glitches.

Update 02/2006: the database is more or less back online. there might be a few glitches though.

Update 05/2004: the virtual crossings have their little circles!

Update 09/2003: finally thru with the revamping... database fully operational.

Update 06/2003: in the process of revamping the database -- one annoying feature, due to new invariants the numbering of the links might be modified. Sorry about that.

An important modification is that the normalization of the Jones polynomials is now such that P(empty)=1. In order to obtain the fairly common normalization P(unknot)=1, you must divide by the polynomial of the unknot. Jones polynomials for tangles have been similarly modified.

2-cabled Jones has now been projected onto the irreducible spin 1 subrep, which is more satisfactory. Spin 3/2 (i.e. four-dimension rep of sl(2)) Jones is also given (why? because I can!)

Remark: the reason that there are only 2 different types of spin 1/2 Jones polynomials for tangles, and not 3 as one might expect for virtual tangles (NW connecting NE, NW connecting SW, but also connecting NW connecting SE with a virtual crossing) is because the last type is forbidden by the alternating character of the tangles.

Update 05/2003: the database is ready and working.

This webpage is part of the project described in the paper
math-ph/0303049

``Matrix Integrals and the Generation and Counting of Virtual Tangles
and Links''

by P. Zinn-Justin and J.-B. Zuber.
Please refer to it for a general overview
of the project.

- Their reduced diagrams encoded as permutations of {0,...,2
*n*-1} (as well as their symmetry factors*s*and their genus*h*). In the case of links, permutations corresponding to mirror images and flips are also given. - Various invariants: number of connected components
*c*, determinant*d*, Jones polynomials (spin 1/2, spin 1, spin 3/2); as well as oriented invariants: linking numbers*l*and extended multi-variable 0th Alexander-Conway polynomial. For the latter lexicographic order is used for the list of orientations and of linking numbers. The connected components are ordered by their labels*i*from 0 to*c*-1 (which are arbitrarily chosen: they might vary from diagram to diagram... similarly which orientation is + and which is - is also arbitrary) and the corresponding variables in the Alexander-Conway polynomial are*t*. In the case of tangles components 0 and 1 are the open components, with the convention that SE external leg belongs to component 0; an additional parameter, the type, describes which leg SE connects to: 1=NW, 2=NE, 3=SW. (note that in our conventions the first crossing of external leg SE is always an overcrossing)._{i}

You can access the database by following this link. Here is a brief summary of how you can use it.

The main tool is the search. Specify various required characteristics of the links you're looking for then hit Search. (note that the "Exact Jones Polynomial" must be given in strict TeX notation with the same ordering as in the database, e.g. "-A^{-2}-A^{2}"). Click on a link number to see a particular link. (Reduced) diagrams are encoded by the permutations σ as in the paper. For compactness σ is encoded in hexadecimal. Click on diagrams for a close up (by modifying the URL you can then play around with the various options of the automated drawing program). You can also display tangles associated to a link (i.e. tangles obtained by removing one crossing from the link diagrams).

Examples: try to search for all links of genus 0 (these are "classical" links) and number of crossings less than 6. Or for all links of genus 1 with trivial Jones polynomial (max deg-min deg=1).

BUGS / TO DO:

- The numbers of morphisms into finite groups are not displayed even though they are mentioned (and used) in the paper.
- The oriented info is missing for links of order 8 and tangles of order 6. Higher spin Jones polynomials are missing for tangles. (these are features, actually, due to storage space limitations)
- The colors of connected components might change from diagram to diagram for a given link/tangle. (the ordering of c.c. is made diagram by diagram as the order of the smallest edge numbers)
- Tangle genus / number of crossing are not provided. (they can be recovered from the associated link, of course)
- It would be nice to somehow visualize virtual links as drawn on thickened surfaces... (now available for genus 1)

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