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.



Bounds for the crossing number in terms of the braid index?
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](https://doi.org/10.4153/CJM-1993-007-x).


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](http://gokovagt.org/proceedings/2008/ggt08-ngthurston.pdf) 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)](https://doi.org/10.1017/S0305004100004576).


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](https://doi:10.1112/jtopol/jtp028), [Studying links via closed braids IV: composite links and split links](https://doi.org/10.1007/BF01233423))


**(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)](https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/topo.12259)


