Problems about the Estimate the Unknotting Number

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.

• I have no idea what "fierce" means here, and I would advise you not to confuse the properties of simple knots like torus knots with properties of arbitrary knots (a typical knot is much, much more complicated [think of a knotted length of rope entirely filling up a room], and guessing patterns is hard). But using the data in indiana.edu/~knotinfo, one can see that your guess is true for all knots with up to 12 crossings. I don't know if it is true in general.
– Sue
Sep 28 '12 at 4:47
• Yes, all knots have unknotting number less than $\lfloor n/2 \rfloor$. Suppose that making $k > \lfloor n/2 \rfloor$ crossing flips results in the unknot, then you can check that flipping all unflipped crossings instead also does. This clearly requires $\leq \lfloor n/2 \rfloor$ flips and so the unknotting number is at most $\lfloor n/2 \rfloor$. An equivalent bound also holds for unlinking links. Sep 28 '12 at 6:00
• @qwerty: Oh, thank you, it seems that I have asked a question that don't make sense, sorry about that Sep 28 '12 at 11:59
• @Sue: Thank you for providing me so useful website! ^^ Sorry about asking the problem that don't make any sense. Sep 28 '12 at 12:04

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 $T_{p,q}$ is $c(T_{p,q})=\min\{(p-1)q,(q-1)p\}$, whereas the unknotting number is $u(T_{p,q})=(p-1)(q-1)/2$ (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 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.