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Murasugi in his book (Knot theory and its applications, page 60) writes:

Conjecture. If $K$ is a knot, then $c(K) \ge 3(br(K) - 1)$, where equality only holds when $K$ is the trivial knot, the trefoil knot, or the (connected) sum of trefoil knots.

I would like to know who came up first with this conjecture, if there was any progress on it in recent years and if it has some special name. Murasugi's book is quite tacit here.

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This is only a partial answer, but perhaps it can point you in the correct direction.

In the paper An estimate of the bridge index of links, Murasugi conjectures that $$3[b(L)−1]\leq c(L)+\mu−1$$ where $b(L)$ is the braid bridge index of $L$, $c(L)$ is the crossing number of $L$, and $\mu$ is the number of components of $L$. In that same paper, he proves the conjecture alternating algebraic links.

I do not have access to the paper, but the MathSciNet review mentions that the conjecture above is due to Fox when $\mu=1$, that is, when $L$ is a knot. The full bibliographic details for the paper are

  • Murasugi, Kunio. An estimate of the bridge index of links. Kobe J. Math. 5 (1988), no. 1, 75–86.
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    $\begingroup$ I also can't access that paper but discovered mathscinet.ams.org/mathscinet-getitem?mr=1200324 which is easier to obtain. The latter points to mathscinet.ams.org/mathscinet-getitem?mr=37510 (Fox, R. H. On the total curvature of some tame knots. Ann. of Math. (2) 52 (1950), 258–260) and this looks indeed like the reference I was looking for. $\endgroup$
    – user471311
    Commented Jul 23, 2020 at 19:02
  • $\begingroup$ I would never ever try looking at older works from Murasugi... nice finding ;) $\endgroup$
    – user471311
    Commented Jul 23, 2020 at 19:02

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