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

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Roughly, this thesis can be divided into three parts. In the first part, we study the Galois conjugates of the dilatation of pseudoAnosov 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 pseudoAnosov 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 fourgenus of positive braid knots. We show that the difference of the ordinary Seifert genus and the topological fourgenus grows at least linearly with the positive braid index. In particular, we show that the positive braid knots for which the topological fourgenus 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:belbes2907 
Additional Information: 
eDissertation (edbe) 
BORIS DOI: 
10.7892/boris.104913 
URI: 
https://boris.unibe.ch/id/eprint/104913 