It is well known that if c(K)=2n+1, then u(K) is less than n+1. It can not be sharper because of the trefoil knot. On the other hand, if c(K)=2n, then similarly we have u(K) is less than n+1. I think u(K)=n is impossible in this case, i.e. there does not exist a knot K with c(K)=2n and u(K)=n. Maybe it is fairly easy, but I have no idea how to deduce it. Any hint is welcome :)
Take the 2minute tour
×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

You can see the answer in Proposition 2.1 of link text 

