Suppose that we have a knot $K$ with 40 crossings which is not a cable knot or an alternating knot.


Then, what is the easiest way to compute Ozsváth-Szabó's invariant $\tau(K)$?


Are there any softwares which compute $\tau(K)$ if we insert Gauss code or PD code of $K$?


For Rasmussen's $s$-invariant, Knottheory package works very nicely for this purpose. 



What is the easiest way to compute Ozsváth-Szabó tau invariant of a knot?
In principle, a computer can compute $\tau$ for any knot, using [grid diagrams](http://msp.org/gt/2007/11-4/p09.xhtml) (or [variants](http://www.worldscientific.com/doi/abs/10.1142/S0218216510007796) [thereof](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=droz_j%2A&s5=floer&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq)); Baldwin and Gillam [computed](http://www.worldscientific.com/doi/abs/10.1142/S0218216512500757) $\tau$ for knots up to 11 crossings a while ago. However, this is not practical. ~~Unlike with Khovanov homology, there are no programs that can compute even $\widehat{HFK}$ for "large" knots (say, more than 20 crossings).~~ I have heard that Zoltán Szabó has a program that can compute $\widehat{HFK}$ for large knots (maybe around 50 crossings) in a matter of minutes; I don't think he has made it public yet, though, and I don't know if it computes $\tau$.


There are a couple of tricks you can use, that sometimes help:


* Plamenevskaya gave a [lower bound](http://msp.org/agt/2004/4-1/p20.xhtml) on $\tau$ using the self-linking number of any representative of $K$; more precisely, if $T$ is a transverse representative of $K$, then
$$ {\rm sl}\,T \le 2\tau(K) - 1.$$
The inequality is in fact an equality if $T$ is the closure of a quasi-positive braid (i.e. the closure of a braid that's a product of conjugates of positive generators).
Of course you can use it for both $K$ and its inverse, and see what you get. To generate some random Legendrian/transverse representatives of $K$ you can use [Gridlink](http://homepages.math.uic.edu/~culler/gridlink/).
* Another lower bound is given in terms of the unknotting number (I believe this is due to Ozsváth and Szabó):
$$ u(K) \le \tau(K),$$
and in fact there are more refined versions of this with signed unknotting numbers (for which I'm afraid I have no references, but I can try and dig them up).
* Of course, you can always try and simplify your knot using concordances, since $\tau$ is a concordance invariant (and then combine it with the previous tricks).
* Upper bounds depend a lot on the kind of information you have. Plamenevskaya's inequality also happens to give an upper bound (since it gives a lower bound on $\tau$ and $-\tau$); the Seifert and slice genus are obvious candidates, but the latter is usually quite hard to get your hands on.


**EDIT**: added something about computing $\widehat{HFK}$ and the relationship between $\tau$ and the self-linking number.


