Bounds for the crossing number in terms of the braid index? Is there a lower bound on the crossing number of a knot (resp., link) with braid index $b$?
For knots, I believe the smallest crossing number for braid index 2 is 3, the smallest crossing number for braid index 3 is 4, the smallest crossing number for braid index 4 is 6, the smallest crossing number for braid index 5 is 8, the smallest crossing number for braid index 6 is 10, and the smallest crossing number for braid index 7 is 12. The urge to extrapolate is strong.
There is no upper bound, of course -- already with braid index 2 there are knots with arbitrarily large (odd) crossing number.
 A: As in the comment of dvitek, as for the relation of the braid index and the crossing number, Ohyama proved $c(L) \geq 2b(L)-2$ in On the Minimal Crossing Number and the Braid Index of Links.
Here I add three additional information.
(a) A simpler proof of $c(L) \geq 2b(L)-2$ (based on other results, but it adds additional insight)
Let $\alpha(L)$ be the arc index of a link. It is not hard to see that
$2b(L) \leq \alpha(L)$
(just try to change a minimum grid diagram into a braid diagram -- see Grid diagrams, braids, and contact geometry for example).
On the other hand, $\alpha(L) \leq c(L)+2$ for a prime link, and equality happens if and only if $L$ is alternating (An upper bound of arc index of links).
Combining these two inequalities we get
$2b(L) \leq c(L)+2$. In particular, when $L$ is prime, the equality occurs if and only if $L$ is alternating and $\alpha(L)=2b(L)$.
(b) Though many results are stated for non-split links, a similar conclusion $2b(L) \leq c(L)+2m$ holds for split links if $L$ is split union $L= L_1\sqcup \cdots \sqcup L_m$ of $m$ non-split links $L$, because $c(L)=c(L_1) + \cdots + c(L_m)$ and $b(L)=b(L_1) + \cdots + b(L_m)$ hold (see The crossing number of composite knots, Studying links via closed braids IV: composite links and split links)
(c)
If you allow to use one additional natural quantity, the maximum euler characteristic $\chi(L)$, we can get lower and upper bounds
$-\chi(L)+b(L) \leq c(L)\leq (2b(L)-5)(-\chi(L)+b(L))$
(A quantitative Birman–Menasco finiteness theorem and its application to crossing number)
