Minimal dilatation in Penner's construction

For all orientable closed surfaces, we determine the minimal dilatation among mapping classes arising from Penner's construction. We also discuss generalisations to surfaces with punctures.Comment: 11 pages, 6 figure

Signature, positive Hopf plumbing and the Coxeter transformation

By a theorem of A'Campo, the eigenvalues of certain Coxeter transformations are positive real or lie on the unit circle. By optimally bounding the signature of tree-like positive Hopf plumbings from below by the genus, we prove that at least two thirds of them lie on the unit circle. In contrast, we show that for divide links, the signature cannot be linearly bounded from below by the genus.Comment: 16 pages, 5 figures, with appendix by Peter Feller and Livio Liecht

Teichm\"uller polynomials of fibered alternating links

We give an algorithm for computing the Teichm\"uller polynomial for a certain class of fibered alternating links associated to trees. Furthermore, we exhibit a mutant pair of such links distinguished by the Teichm\"uller polynomial.Comment: 22 pages, 12 figure

Divide knots of maximal genus defect

We construct divide knots with arbitrary smooth four-genus but topological four-genus equal to one. In particular, for strongly quasipositive fibred knots, the ratio between the topological and the smooth four-genus can be arbitrarily close to zero.Comment: 8 pages, 3 figure

Signature and concordance of positive knots

We derive a linear estimate of the signature of positive knots, in terms of their genus. As an application, we show that every knot concordance class contains at most finitely many positive knots.Comment: 10 pages, 7 figure