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For an alternating knot K, checkerboard-color the knot (if this is a lousy ASCII crossing: %, white goes to the left/right and black to top/bottom). Assume no surplus Reidemeister 1 kinks exist (K has minimal crossing number =C). Call black-white areas D and the writhe W. Then for the Thurston-Bennequin numbers of K and mirror(K) (call them X and Y) the following equations hold (modulo sign error and typo :-):
$ C=-X-Y-2;D+2*W=Y-X$
Surely this is known?!
The tricky part comes with nonalternating knots. I played around with minimal crossing number representants of knots, and with proper tweaking the second equation still seems to hold, while for the first, the difference somehow seems to be connected with Stasiaks "natural order of knots". I could conjecture a lot :-) but what is actually known about generalizing these equations to nonalternating knots? Oh, and does somebody have the T-B N for small links? E.g. for the link 4_2 I would assume them to be -5 and -1.

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1 Answer 1

up vote 4 down vote accepted

The first identity you conjecture is true only for alternating knots.

The identity for alternating knots is mentioned in one of Lenny Ng's papers (at the end of page 3), where he also gives references and a sketch of the proof.

In the same paper, he reproves a result of Matsuda (see here), which is an inequality relating the arc index $\alpha$ of a knot and the maximal Thurston-Bennequin numbers of the knot and its mirror; combining this with a result of Bae and Park (see here) relating $\alpha$ with the crossing number $c$, one can show that for every non-alternating knot $K$ there's an inequality $2+c(K)+\overline{tb}(K)+\overline{tb}(m(K)) > 0$.

Finally, if you want to know stuff about Legendrian representatives of knots and links with few crossings, Lenny Ng's homepage is a place you have to check (in particular his atlases, that are joint work with Chongchitmate).

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THX, that answer was spot-on. I check out the paper. –  Hauke Reddmann Oct 5 '12 at 10:26
    
For the record, there's some assumption about the knots involved being prime that I didn't mention: see the remarks at the end of Bae and Park's paper. –  Marco Golla Oct 5 '12 at 12:04

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