POSITIVE PRESENTATIONS OF SURFACE BRAID GROUPS

2007 ◽  
Vol 16 (09) ◽  
pp. 1219-1233 ◽  
Author(s):  
PAOLO BELLINGERI ◽  
EDDY GODELLE
Keyword(s):  
Conjugacy Problem ◽  
Braid Groups ◽  
Group Presentations ◽  
Positive Presentations

We provide new group presentations for surface braid groups which are positive. We study some properties of such presentations and we solve the conjugacy problem in a particular case.

scholarly journals Holonomic and Legendrian parametrizations of knots

2000 ◽  
Vol 09 (03) ◽  
pp. 293-309 ◽  
Author(s):  
Joan S. Birman ◽  
Nancy C. Wrinkle
Keyword(s):  
Conjugacy Problem ◽  
Braid Groups ◽  
Algebraic Solution ◽  
Legendrian Knots

Holonomic parametrizations of knots were introduced in 1997 by Vassiliev, who proved that every knot type can be given a holonomic parametrization. Our main result is that any two holonomic knots which represent the same knot type are isotopic in the space of holonomic knots. A second result emerges through the techniques used to prove the main result: strong and unexpected connections between the topology of knots and the algebraic solution to the conjugacy problem in the braid groups, via the work of Garside. We also discuss related parametrizations of Legendrian knots, and uncover connections between the concepts of holonomic and Legendrian parametrizations of knots.

ABELIAN AND METABELIAN QUOTIENT GROUPS OF SURFACE BRAID GROUPS

2016 ◽  
Vol 59 (1) ◽  
pp. 119-142 ◽  
Author(s):  
PAOLO BELLINGERI ◽  
EDDY GODELLE ◽  
JOHN GUASCHI
Keyword(s):  
Braid Groups ◽  
Rigidity Results ◽  
Group Presentations ◽  
Quotient Groups ◽  
Exact Sequences

AbstractIn this paper, we study Abelian and metabelian quotients of braid groups of oriented surfaces with boundary components. We provide group presentations and we prove rigidity results for these quotients arising from exact sequences related to (generalised) Fadell–Neuwirth fibrations.

A Note on the Shifted Conjugacy Problem in Braid Groups

2009 ◽  
Vol 1 (2) ◽  
Author(s):  
Arkadius Kalka ◽  
Eran Liberman ◽  
Mina Teicher
Keyword(s):  
Conjugacy Problem ◽  
Braid Groups

scholarly journals Term Rewriting for the Conjugacy Problem and the Braid Groups

1994 ◽  
Vol 18 (6) ◽  
pp. 563-572 ◽  
Author(s):  
John Pedersen ◽  
Margaret Yoder
Keyword(s):  
Conjugacy Problem ◽  
Term Rewriting ◽  
Braid Groups

scholarly journals Garside theory and subsurfaces: Some examples in braid groups

2019 ◽  
Vol 11 (2) ◽  
pp. 61-75
Author(s):  
Saul Schleimer ◽  
Bert Wiest
Keyword(s):  
Conjugacy Class ◽  
Geometric Property ◽  
Conjugacy Problem ◽  
Fixed Number ◽  
Braid Groups ◽  
Worst Case ◽  
Curve Graph ◽  
Precise Bound

Abstract Garside-theoretical solutions to the conjugacy problem in braid groups depend on the determination of a characteristic subset of the conjugacy class of any given braid, e.g. the sliding circuit set. It is conjectured that, among rigid braids with a fixed number of strands, the size of this set is bounded by a polynomial in the length of the braids. In this paper we suggest a more precise bound: for rigid braids with N strands and of Garside length L, the sliding circuit set should have at most {C\cdot L^{N-2}} elements, for some constant C. We construct a family of braids which realise this potential worst case. Our example braids suggest that having a large sliding circuit set is a geometric property of braids, as our examples have multiple subsurfaces with large subsurface projection; thus they are “almost reducible” in multiple ways, and act on the curve graph with small translation distance.

scholarly journals Conjugacy problem for braid groups and Garside groups

2003 ◽  
Vol 266 (1) ◽  
pp. 112-132 ◽  
Author(s):  
Nuno Franco ◽  
Juan González-Meneses
Keyword(s):  
Conjugacy Problem ◽  
Braid Groups

Pseudo-Anosov Theory

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Braid Groups ◽  
Intersection Numbers ◽  
Branched Covers ◽  
Algebraic Integers ◽  
Dehn Twists ◽  
Mapping Classes ◽  
Dynamical Properties

This chapter focuses on the construction as well as the algebraic and dynamical properties of pseudo-Anosov homeomorphisms. It first presents five different constructions of pseudo-Anosov mapping classes: branched covers, constructions via Dehn twists, homological criterion, Kra's construction, and a construction for braid groups. It then proves a few fundamental facts concerning stretch factors of pseudo-Anosov homeomorphisms, focusing on the theorem that pseudo-Anosov stretch factors are algebraic integers. It also considers the spectrum of pseudo-Anosov stretch factors, along with the special properties of those measured foliations that are the stable (or unstable) foliations of some pseudo-Anosov homeomorphism. Finally, it describes the orbits of a pseudo-Anosov homeomorphism as well as lengths of curves and intersection numbers under iteration.

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