On the spectra of mapping classes and the 4-genera of positive knots

Liechti, Nicola Livio (2017). On the spectra of mapping classes and the 4-genera of positive knots. (Dissertation, Universität Bern, Philosophisch-naturwissenschaftliche Fakultät)

[img]
Preview
Text
17liechti_l.pdf - Published Version
Available under License Creative Commons: Attribution-Noncommercial-No Derivative Works (CC-BY-NC-ND).

Download (3MB) | Preview

Roughly, this thesis can be divided into three parts. In the first part, we study the Galois conjugates of the dilatation of pseudo-Anosov mapping classes. In particular, for a product of two multitwists, we show that all Galois conjugates are either real and positive or contained in the unit circle and the positive real axis, depending on whether the products are of opposite or of the same sign. Furthermore, for each closed orientable surface, we determine the minimal dilatation among pseudo-Anosov mapping classes arising from Penner's construction. In the second part, we consider the Alexander polynomial and the signature function of links. For a Murasugi sum of two Seifert surfaces with symmetric, definite Seifert form, we show that all zeroes of the Alexander polynomial are either real and positive or contained in unit circle and the negative real axis, depending on whether the Seifert forms are definite of opposite or the same sign. Furthermore, we prove that the signature function of a Murasugi sum of two Seifert surfaces with symmetric, definite Seifert form is monotonic. We also show that the signature of a positive arborescent Hopf plumbing is greater than or equal to two thirds of the first Betti number. In the third part, we study the topological four-genus of positive braid knots. We show that the difference of the ordinary Seifert genus and the topological four-genus grows at least linearly with the positive braid index. In particular, we show that the positive braid knots for which the topological four-genus equals the ordinary Seifert genus are exactly the positive braid knots with maximal signature invariant.

Item Type:

Thesis (Dissertation)

Division/Institute:

08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematics

UniBE Contributor:

Liechti, Nicola Livio and Baader, Sebastian

Subjects:

500 Science > 510 Mathematics

Language:

English

Submitter:

Igor Hammer

Date Deposited:

15 Aug 2017 11:50

Last Modified:

15 Aug 2017 11:50

URN:

urn:nbn:ch:bel-bes-2907

Additional Information:

e-Dissertation (edbe)

BORIS DOI:

10.7892/boris.104913

URI:

https://boris.unibe.ch/id/eprint/104913

Actions (login required)

Edit item Edit item
Provide Feedback