scholarly journals THE CULLER-SHALEN SEMINORMS OF THE (-2, 3, n) PRETZEL KNOT

2002 ◽  
Vol 11 (08) ◽  
pp. 1251-1289 ◽  
Author(s):  
THOMAS W. MATTMAN
Keyword(s):  
Fundamental Group ◽  
Algebraic Curves ◽  
Newton Polygon ◽  
Explicit Calculation ◽  
Character Variety ◽  
Fundamental Polygon

We show that the [Formula: see text]-character variety of the (-2, 3, n) pretzel knot consists of two (respectively three) algebraic curves when 3 ∤ n (respectively 3 | n) and given an explicit calculation of the Culler-Shalem seminorms of these curves. Using this calculation, we describe the fundamental polygon and Newton polygon for these knots and give a list of Dehn surgerise yielding a manifold with finite or cyclic fundamental group. This constitutes a new proof of property P for these knots.

Various Generalizations and Deformations of PSL(2,ℝ) Surface Group Representations and their Higgs Bundles

2018 ◽  
pp. 471-502
Author(s):  
Brian Collier
Keyword(s):  
Fundamental Group ◽  
Vector Bundles ◽  
Surface Group ◽  
Closed Surface ◽  
Connected Components ◽  
Character Variety ◽  
Higgs Bundles ◽  
Group Representations ◽  
Symmetric Powers ◽  
Higher Rank

The goal of this chapter is to examine the various ways in which Fuchsian representations of the fundamental group of a closed surface of genus g into PSL(2, R) and their associated Higgs bundles generalize to the higher-rank groups PSL(n, R), PSp(2n, R), SO0(2, n), SO0(n,n+1) and PU(n, n). For the SO0(n,n+1)-character variety, it parameterises n(2g−2) new connected components as the total spaces of vector bundles over appropriate symmetric powers of the surface, and shows how these components deform in the character variety. This generalizes results of Hitchin for PSL(2, R).

scholarly journals Frobenius algebras derived from the Kauffman bracket skein algebra

2016 ◽  
Vol 25 (04) ◽  
pp. 1650016 ◽  
Author(s):  
Charles Frohman ◽  
Nel Abdiel
Keyword(s):  
Fundamental Group ◽  
Central Extension ◽  
Function Field ◽  
Underlying Surface ◽  
Character Variety ◽  
Oriented Surface ◽  
Variable Formula ◽  
Kauffman Bracket ◽  
Frobenius Algebras ◽  
Character Ring

The Kauffman bracket skein algebra of a compact oriented surface when the variable [Formula: see text] in the Kauffman bracket is set equal to [Formula: see text], where [Formula: see text] is an odd counting number, is a central extension of the ring of [Formula: see text]-characters of the fundamental group of the underlying surface. In this paper, we construct symmetric Frobenius algebras from the Kauffman bracket skein algebra of some simple surfaces by two strategies. The first is to localize the skein algebra at the characters so it becomes an algebra over the function field of the character variety of the surface, and the second is to specialize at a place of the character ring.

scholarly journals PUISEUX POWER SERIES SOLUTIONS FOR SYSTEMS OF EQUATIONS

2010 ◽  
Vol 21 (11) ◽  
pp. 1439-1459 ◽  
Author(s):  
FUENSANTA AROCA ◽  
GIOVANNA ILARDI ◽  
LUCÍA LÓPEZ DE MEDRANO
Keyword(s):  
Algebraic Curves ◽  
Newton Polygon ◽  
Series Solutions ◽  
Systems Of Equations ◽  
Algebraic Equations ◽  
Puiseux Series ◽  
Tropical Variety ◽  
Power Series Solutions ◽  
Tropical Varieties ◽  
The Ideal

We give an algorithm to compute term-by-term multivariate Puiseux series expansions of series arising as local parametrizations of zeroes of systems of algebraic equations at singular points. The algorithm is an extension of Newton's method for plane algebraic curves, replacing the Newton polygon by the tropical variety of the ideal generated by the system. As a corollary we deduce a property of tropical varieties of quasi-ordinary singularities.

REMARKS ON THE A-POLYNOMIAL OF A KNOT

1996 ◽  
Vol 05 (05) ◽  
pp. 609-628 ◽  
Author(s):  
D. COOPER ◽  
D.D. LONG
Keyword(s):  
Fundamental Group ◽  
Newton Polygon ◽  
Polynomial Invariant ◽  
Incompressible Surfaces ◽  
Open Questions ◽  
Boundary Slopes

This paper reviews the two variable polynomial invariant of knots defined using representations of the fundamental group of the knot complement into [Formula: see text]. The slopes of the sides of the Newton polygon of this polynomial are boundary slopes of incompressible surfaces in the knot complement. The polynomial also contains information about which surgeries are cyclic, and about the shape of the cusp when the knot is hyperbolic. We prove that at least some mutants have the same polynomial, and that most untwisted doubles have non-trivial polynomial. We include several open questions.

Filling words in fundamental groups of Riemann surfaces

2013 ◽  
Vol 50 (1) ◽  
pp. 31-50
Author(s):  
C. Zhang
Keyword(s):  
Riemann Surface ◽  
Fundamental Group ◽  
Riemann Surfaces ◽  
Fundamental Groups ◽  
Closed Geodesics ◽  
New Words

The purpose of this article is to utilize some exiting words in the fundamental group of a Riemann surface to acquire new words that are represented by filling closed geodesics.

Representations of the fundamental group and the torsor of deformations. Global aspects

2017 ◽  
Author(s):  
Ahmed Abbes ◽  
Michel Gros
Keyword(s):  
Fundamental Group ◽  
Koszul Complex ◽  
Finiteness Conditions ◽  
Inverse Systems ◽  
Logarithmic Geometry

This chapter continues the construction and study of the p-adic Simpson correspondence and presents the global aspects of the theory of representations of the fundamental group and the torsor of deformations. After fixing the notation and general conventions, the chapter develops preliminaries and then introduces the results and complements on the notion of locally irreducible schemes. It also fixes the logarithmic geometry setting of the constructions and considers a number of results on the Koszul complex. Finally, it develops the formalism of additive categories up to isogeny and describes the inverse systems of a Faltings ringed topos, with a particular focus on the notion of adic modules and the finiteness conditions adapted to this setting. The chapter rounds up the discussion with sections on Higgs–Tate algebras and Dolbeault modules.

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