A conjecture formulated and popularized by the reviewer states that the signature of a positive link is bounded increasingly by its (negated) Euler characteristic (or Betti number, as used here). This paper proves (Theorem 1) such a linear lower bound for tree-like positive Hopf plumbing links (which are obviously positive). In contrast, it is shown that for A’Campo divide links (not necessarily positive, as follows from the remarks below), the signature cannot be linearly bounded from below by the genus. However, for slalom knots there is again a linear lower bound (Theorem 4).
There are several developments on the topics treated here that must be mentioned.
Baader-Dehornoy-Liechti have fully resolved the Increasing Bound Conjecture; in fact, they proved a linear lower bound for arbitrary positive links. Feller had previously given a linear lower bound for the positive braid links (for which a cube-root-like bound was known from the reviewer’s old work).
The result of A’Campo quoted in the Introduction has been extended to tree-like plumbing links for generalized Hopf bands (any positive number of full twists) by Hirasawa-Murasugi and independently (and more simply) by Baader and the reviewer (to appear).
Section 5 discusses Hoste’s conjecture on the zeros of the Alexander polynomial of an alternating knot or link and proposes an analogous conjecture (Conjecture 13) for a positive braid link. In the reviewer’s minor modification (and strengthening) it states this: if \(L\) is a positive braid link of \(n\) components, then any zero of the reduced Alexander polynomial \(\Delta_L(t)/(t^{1/2}-t^{-1/2})^{n-1}\) has real part smaller than \(1\). This conjecture may potentially trigger some intriguing piece of research. The reviewer’s experiments with low-crossing knots suggest that it may be true even for positive links. The aforementioned work with Baader has confirmed the conjecture for tree-like plumbing links of generalized Hopf bands (any positive number of full twists).
An Appendix (joint with Feller) contains a proof that the absolute value of the signature of a link is a lower bound for the number of zeros of the Alexander polynomial lying on the unit circle. This fact was known in special cases but was plagued by a painful (recording) history. Some more explanation may be found in a paper of P. M. Gilmer and C. Livingston [Proc. Am. Math. Soc. 144, No. 12, 5407–5417 (2016; Zbl 1361.57010)], who have given an alternative proof. They also point out there is a minor oversight in the present paper (concerning a signature jump at \(-1\)), which can be easily fixed, though.