For the definition of unknotting Number, you can assess <http://www.popmath.org.uk/exhib/pagesexhib/unknum.html>


My question is:


For given a knot K, let n be the crossing number of K, is their any estimate of the unknotting number of K? Of course, the unknotting number is smaller than n-1, but does there exist any nontrivial estimate? 


I guess the unknotting number may be smaller than $[\frac{n}{2}]$. Based on the fact that the unknotting number of the torus knot is no bigger than $[\frac{n}{2}]$. I think the torus knot might "tight" in the most "fierce" way. 



Problems about the Estimate the Unknotting Number 
You're intuition is right: asymptotically, the torus knots have unknotting number roughly half of the crossing number. The [crossing number of a $(p,q)$ torus knot](http://en.wikipedia.org/wiki/Torus_knot) $T\_{p,q}$ is $c(T\_{p,q})=\min\{(p-1)q,(q-1)p\}$, whereas the [unknotting number is $u(T\_{p,q})=(p-1)(q-1)/2$](http://en.wikipedia.org/wiki/Unknotting_number) (this follows from the fact that the 4-ball genus gives a lower bound on the unknotting number, and the solution to the [Milnor conjecture](http://en.wikipedia.org/wiki/Milnor_conjecture_%28topology%29) by Kronheimer-Mrowka). So asymptotically, the ratio $u(T\_{p,q})/c(T\_{p,q}) \to \frac12$ , as $p,q \to \infty$, realizing the maximal possible ratio. 


