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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.

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  • $\begingroup$ The inequality $c(L) \ge 2b(L)-2$ for non-split links was proved in Ohyama, "On the minimal crossing number and the braid index of links" (1993), which (for me) is the first result on Google Scholar for the query [crossing number braid index]. For the knot case, if $b(K) = 2$ showing that $c(K) \ge 3$ is trivial. I do not know (a) how to establish equality or (b) whether there are any subtleties in the split case. $\endgroup$
    – dvitek
    Commented Feb 16, 2023 at 20:21

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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)

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  • $\begingroup$ Is it sharp? That is, is it true that for every $n>2$ there is a knot $L$ with $b(L)=n$ and $c(L)=2b(L)-2=2n-2$? $\endgroup$
    – Charles
    Commented Feb 17, 2023 at 18:00
  • $\begingroup$ Thanks for clarifying the split-link case; for some reason I was concerned about the braid index formula but the lower bound is in fact straightforward. (I think your link #4 to Lackenby's crossing-number paper is broken.) $\endgroup$
    – dvitek
    Commented Feb 17, 2023 at 19:13
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    $\begingroup$ @Charles The 2n-twsit knot K_n (n=1 case is the Figure-eight) attains b(K_n)=n+2 and c(K_n)=2n+2. Also, the connected sum of (k-1) Hopf links L_k attains b(L_k)=k+1 and c(L_k)=2k. The (2,2n)-torus link with opposite orientations of two strands also gives an example of 2-component link where the equality is strict. $\endgroup$
    – Tetsuya Ito
    Commented Feb 20, 2023 at 1:02

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