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Orderability theory for braid groups over surfaces and for Link-Homotopy generalized String-Links over surfaces. Representation theorem for the Link-Homotopy generalized String-Links over surfaces

Grant number: 14/08088-7
Support type:Scholarships in Brazil - Post-Doctorate
Effective date (Start): November 01, 2014
Effective date (End): October 31, 2016
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Geometry and Topology
Cooperation agreement: Coordination of Improvement of Higher Education Personnel (CAPES)
Principal Investigator:Denise de Mattos
Grantee:Juliana Roberta Theodoro de Lima
Home Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
Associated scholarship(s):14/27135-6 - Orderability theory for braid groups over surfaces and for link-homotopy generalized string-links over surfaces, BE.EP.PD

Abstract

In 1925, Artin introduced the study of braid groups, which has relation with knots and links theory. There are two important results obtained by Artin, namely, the presentation and representation theorem for braid groups: the first one, give us a presentation for the braid groups over the unitary disk, that consists basically to cognize this group through its generators and relations between its elements. The second result, so much important as the first one, gives us an isomorphism between the braid groups over the unitary disk on n strands and a subgroup of a free group with rank n.The theory of braids had developed in many directions with works of Alexander, Markov, Birman, Rolfsen, Paris, Gonzalez-Meneses, Bellingeri, Boyer and others.Recently, an important area that has been developed is called braid groups on surfaces with genus: the presentation theorem for the braid groups on surfaces, is an generalization for the presentation for the braid groups over the disk made by Artin, was made byGonzales-Meneses. He founded the smallest presentation for braid groups over surfaces. In this paper, he produces the presentation for the braid groups over orientable surfaces of genus g more or equal than 1 and for the non orientable surfaces of genus g more or equal than 2. In particular, this paper was de central paper of study in the master's degree of the mentioned student.Also, the representation theorem for the braid groups over surfaces was founded to by Bardakov and Belingeri, generalizing the representation theorem over the disk made by Artin. Now, for the orderability theory for groups, Rolfsen, Dynnikov, Dehornoy and West had founded that the braid groups over the unitary disk is left-orderable, i.e, there is a left, strict and linear ordering for this group. Also, they showed the pure braid groups over the unitary disk is bi-orderable, i.e., there is a left and right, strict and linear ordering for this group. All details about this result can be found in [R]. Later, Gonzalez-Meses proved that the pure braid groups over surfaces is bi-orderable.Further, we can talk about the generalized string-links. Informally, we say that a generalized string-link is a generalization of a braid. For the first one, we consider the link-homotopy equivalence relation and for the second one, we consider the isotopy equivalence relation. We say that generalized string links are generalizations of braids since link-homotopy is a finite numbers os isotopies and crossing changes.One os the properties of link-homotopy is to allow that each string in a generalized string-link has self intersection, i.e., to allow that it has a finite number of self crossings (crossing changes). The generalized string-links with this last equivalence relation, provided the concatenation, became a group, called Homotopy Generalized String Links Over Surfaces. Ekaterina proved in her phd's thesis, advised by Rolfsen, that Homotopy String Links Over the disk is bi-orderable. Note that when we say only "string-links" we refer the pure case of generalized string-links. Now, for this proposal, we are considering another definition: we say that a generalized string-link is of second type when we allow that the self interseccions of a given generalized string-link was substituted by trivial segment.In this research proposal, we purpose develop generalizations for the results mentioned above. More specifically, we have as aim the approach of the following problems related with this theme:1. Representation Theorem for the Link-Homotopy Group of Generalized String-Links on Surfaces.2. Study the conditions to the set of generalized string links of second type equipped with the operation of concatenation become a well defined group and find a presentation and a representation theorem for such group.3. Prove that the Braid Group on Surfaces is left-orderable.4. Prove that the Link-Homotopy Group of Generalized String Link on Surfaces is left-orderable. (AU)

Scientific publications
(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)
THEODORO DE LIMA, JULIANA R.; DE MATTOS, DENISE. Ordering homotopy string links over surfaces. JOURNAL OF KNOT THEORY AND ITS RAMIFICATIONS, v. 25, n. 1 JAN 2016. Web of Science Citations: 0.

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