arXiv subjects

Livio Liechti

Publications and source records attributed to Livio Liechti.

25 records · Page 2Linked to original sources

arXiv2019

Divide monodromies and antitwists on surfaces

A divide on an orientable 2-orbifold gives rise to a fibration of the unit tangent bundle to the orbifold.We characterize the corresponding monodromies as exactly the products of a left-veering horizontal and a right-veering vertical antitwist with respect to a cylinder decomposition, where the notion of an antitwist is an orientation-reversing analogue of a multitwist. Many divide monodromies are pseudo-Anosov and we give plenty of examples.In particular, we show that there exist divide monodromies with stretch factor arbitrarily close to one, and give an example none of whose powers can be obtained by Penner's or Thurston's construction of pseudo-Anosov mapping classes.As a side product, we also get a new combinatorial construction of pseudo-Anosov mapping classes in terms of products of antitwists.

math.DS↗

arXiv2018

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.

math.GT↗

arXiv2017

Positive braid knots of maximal topological 4-genus

We show that a positive braid knot has maximal topological 4-genus exactly if it has maximal signature invariant. As an application, we determine all positive braid knots with maximal topological 4-genus and compute the topological 4-genus for all positive braid knots with up to 12 crossings.

math.GT↗

arXiv2017

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.

math.GT↗

arXiv2016

On the topological 4-genus of torus knots

We prove that the topological locally flat slice genus of large torus knots takes up less than three quarters of the ordinary genus. As an application, we derive the best possible linear estimate of the topological slice genus for torus knots with non-maximal signature invariant.

math.GT↗

arXiv2016

On Coxeter mapping classes and fibered alternating links

Alternating-sign Hopf plumbing along a tree yields fibered alternating links whose homological monodromy is, up to a sign, conjugate to some alternating-sign Coxeter transformation. Exploiting this tie, we obtain results about the location of zeros of the Alexander polynomial of the fibered link complement implying a strong case of Hoste's conjecture, the trapezoidal conjecture, bi-orderability of the link group, and a sharp lower bound for the homological dilatation of the monodromy of the fibration. The results extend to more general hyperbolic fibered 3-manifolds associated to alternating-sign Coxeter graphs.

math.GT↗

arXiv2016

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.

math.GT↗