scholarly journals Character varieties of even classical pretzel knots

2019 ◽  
Vol 56 (4) ◽  
pp. 510-522
Author(s):  
Haimiao Chen
Keyword(s):  
Character Variety ◽  
Character Varieties ◽  
Pretzel Knots

Abstract For each even classical pretzel knot P(2k1 + 1, 2k2 + 1, 2k3), we determine the character variety of irreducible SL (2, ℂ)-representations, and clarify the steps of computing its A-polynomial.

scholarly journals Character varieties of odd classical pretzel knots

2018 ◽  
Vol 29 (09) ◽  
pp. 1850060 ◽  
Author(s):  
Haimiao Chen
Keyword(s):  
Character Variety ◽  
Character Varieties ◽  
Pretzel Knots

We determine the [Formula: see text]-character variety for each odd classical pretzel knot [Formula: see text], and present a method for computing its A-polynomial.

Toric Geometry ofSL2(ℂ) Free Group Character Varieties from Outer Space

2018 ◽  
Vol 70 (2) ◽  
pp. 354-399 ◽  
Author(s):  
Christopher Manon
Keyword(s):  
Free Group ◽  
Outer Space ◽  
Integrable Hamiltonian System ◽  
Outer Automorphism ◽  
Character Variety ◽  
Toric Geometry ◽  
Character Varieties ◽  
Simplicial Space ◽  
Proper Map ◽  
Divisorial Valuations

AbstractCuller and Vogtmann defined a simplicial spaceO(g), calledouter space, to study the outer automorphism group of the free groupFg. Using representation theoretic methods, we give an embedding ofO(g) into the analytification of X(Fg,SL2(ℂ)), theSL2(ℂ) character variety ofFg, reproving a result of Morgan and Shalen. Then we show that every pointvcontained in a maximal cell ofO(g) defines a flat degeneration of X(Fg,SL2(ℂ)) to a toric varietyX(PΓ). We relate X(Fg,SL2(ℂ)) andX(v) topologically by showing that there is a surjective, continuous, proper map Ξv:X(Fg,SL2(ℂ)) →X(v). We then show that this map is a symplectomorphism on a dense open subset of X(Fg, SL2(ℂ)) with respect to natural symplectic structures on X(Fg, SL2(ℂ)) andX(v). In this way, we construct an integrable Hamiltonian system in X(Fg, SL2(ℂ)) for each point in a maximal cell ofO(g), and we show that eachvdefines a topological decomposition of X(Fg, SL2(ℂ)) derived from the decomposition ofX(PΓ) by its torus orbits. Finally, we show that the valuations coming from the closure of a maximal cell inO(g) all arise as divisorial valuations built from an associated projective compactification of X(Fg, SL2(ℂ)).

scholarly journals The mapping class group action on -character varieties

2020 ◽  
pp. 1-15
Author(s):  
WILLIAM M. GOLDMAN ◽  
SEAN LAWTON ◽  
EUGENE Z. XIA
Keyword(s):  
Group Action ◽  
Mapping Class Group ◽  
Boundary Component ◽  
Class Group ◽  
Orientable Surface ◽  
Mapping Class ◽  
Character Variety ◽  
Character Varieties ◽  
Compact Orientable Surface

Let $\unicode[STIX]{x1D6F4}$ be a compact orientable surface of genus $g=1$ with $n=1$ boundary component. The mapping class group $\unicode[STIX]{x1D6E4}$ of $\unicode[STIX]{x1D6F4}$ acts on the $\mathsf{SU}(3)$ -character variety of $\unicode[STIX]{x1D6F4}$ . We show that the action is ergodic with respect to the natural symplectic measure on the character variety.

scholarly journals TWISTED ALEXANDER POLYNOMIALS AND CHARACTER VARIETIES OF 2-BRIDGE KNOT GROUPS

2012 ◽  
Vol 23 (06) ◽  
pp. 1250022 ◽  
Author(s):  
TAEHEE KIM ◽  
TAKAYUKI MORIFUJI
Keyword(s):  
Alexander Polynomial ◽  
Character Variety ◽  
Character Varieties ◽  
Alexander Polynomials ◽  
Twisted Alexander Polynomial

We study the twisted Alexander polynomial from the viewpoint of the SL (2, ℂ)-character variety of nonabelian representations of a knot group. It is known that if a knot is fibered, then the twisted Alexander polynomials associated with nonabelian SL (2, ℂ)-representations are all monic. In this paper, we show that for a 2-bridge knot there exists a curve component in the SL (2, ℂ)-character variety such that if the knot is not fibered then there are only finitely many characters in the component for which the associated twisted Alexander polynomials are monic. We also show that for a 2-bridge knot of genus g, in the above curve component for all but finitely many characters the associated twisted Alexander polynomials have degree 4g - 2.

scholarly journals Poincaré series of character varieties for nilpotent groups

2019 ◽  
Vol 22 (3) ◽  
pp. 419-440 ◽  
Author(s):  
Mentor Stafa
Keyword(s):  
Lie Group ◽  
Free Monoid ◽  
Nilpotent Groups ◽  
Poincaré Series ◽  
Character Variety ◽  
Finitely Generated ◽  
Trivial Representation ◽  
Poincare Series ◽  
Character Varieties ◽  
Path Component

Abstract For any compact, connected Lie group G and any finitely generated nilpotent group Γ, we determine the cohomology of the path component of the trivial representation of the group character variety (representation space) {{\rm Rep}(\Gamma,G)_{1}} , with coefficients in a field {{\mathbb{F}}} with characteristic 0 or relatively prime to the order of the Weyl group W. We give explicit formulas for the Poincaré series. In addition, we study G-equivariant stable decompositions of subspaces {{\rm X}(q,G)} of the free monoid {J(G)} generated by the Lie group G, obtained from representations of finitely generated free nilpotent groups.

scholarly journals Traceless SU(2) representations of 2-stranded tangles

2016 ◽  
Vol 162 (1) ◽  
pp. 101-129 ◽  
Author(s):  
YOSHIHIRO FUKUMOTO ◽  
PAUL KIRK ◽  
JUANITA PINZÓN-CAICEDO
Keyword(s):  
Conjugacy Classes ◽  
Character Variety ◽  
Restriction Map ◽  
Pretzel Knots ◽  
Non Linear

AbstractGiven a 2-stranded tangle T contained in a ℤ-homology ball Y, we investigate the character variety R(Y, T) of conjugacy classes of traceless SU(2) representations of π1(Y \ T). In particular we completely determine the subspace of binary dihedral representations, and identify all of R(Y, T) for many tangles naturally associated to knots in S3. Moreover, we determine the image of the restriction map from R(T, Y) to the traceless SU(2) character variety of the 4-punctured 2-sphere (the pillowcase). We give examples to show this image can be non-linear in general, and show it is linear for tangles associated to pretzel knots.

scholarly journals TWISTED ALEXANDER POLYNOMIALS ON CURVES IN CHARACTER VARIETIES OF KNOT GROUPS

2013 ◽  
Vol 24 (03) ◽  
pp. 1350022 ◽  
Author(s):  
TAEHEE KIM ◽  
TAKAHIRO KITAYAMA ◽  
TAKAYUKI MORIFUJI
Keyword(s):  
Alexander Polynomial ◽  
Character Variety ◽  
Character Varieties ◽  
Alexander Polynomials ◽  
Twisted Alexander Polynomial

For a fibered knot in the 3-sphere the twisted Alexander polynomial associated to an SL(2, ℂ)-character is known to be monic. It is conjectured that for a nonfibered knot there is a curve component of the SL(2, ℂ)-character variety containing only finitely many characters whose twisted Alexander polynomials are monic, i.e. finiteness of such characters detects fiberedness of knots. In this paper, we discuss the existence of a certain curve component which relates to the conjecture when knots have nonmonic Alexander polynomials. We also discuss the similar problem of detecting the knot genus.

scholarly journals Algebras of Quantum Monodromy Data and Character Varieties

2018 ◽  
pp. 39-68 ◽  
Author(s):  
Leonid Chekhov ◽  
Marta Mazzocco ◽  
Vladimir Rubtsov
Keyword(s):  
Riemann Surface ◽  
Poisson Structure ◽  
Character Variety ◽  
Representation Variety ◽  
Character Varieties ◽  
Fundamental Groupoid

This chapter examines the Poisson structure of the representation variety of the fundamental groupoid of a Riemann surface with punctures and cusps, and the associated decorated character variety.

CHARACTER VARIETIES AND PERIPHERAL POLYNOMIALS OF A CLASS OF KNOTS

2003 ◽  
Vol 12 (08) ◽  
pp. 1093-1130 ◽  
Author(s):  
HUGH M. HILDEN ◽  
MARÍA TERESA LOZANO ◽  
JOSÉ MARÍA MONTESINOS AMILIBIA
Keyword(s):  
Character Variety ◽  
Hyperbolic Structures ◽  
Character Varieties

For a hyperbolic knot, the excellent curve is the union of components of the character variety containing the characters of the complete hyperbolic structures on the complement of the knot. The peripheral polynomials define the excellent curve in terms of traces of bases of the boundary torus around the knot. In this paper we introduce the concept of net presentation of a knot with 2n strings which generalises the well known plat presentation of a knot with 2n strings. Nets with 4 strings are the correct setting to apply quaternion methods to compute the excellent curve. This is done here in a convenient way for application to the study of invariants of cone-manifold structures.

scholarly journals On the traceless SU(2) character variety of the 6-punctured 2-sphere

2017 ◽  
Vol 26 (02) ◽  
pp. 1740009
Author(s):  
Paul Kirk
Keyword(s):  
Singular Points ◽  
Character Variety ◽  
Kummer Surface ◽  
Branch Locus ◽  
Character Varieties ◽  
Branched Cover

We exhibit the traceless SU(2) character variety of a 6-punctured 2-sphere as a 2-fold branched cover of [Formula: see text], branched over the singular Kummer surface, with the branch locus in [Formula: see text] corresponding to the binary dihedral representations. This follows from an analysis of the map induced on SU(2) character varieties by the 2-fold branched cover [Formula: see text] branched over [Formula: see text] points, combined with the theorem of Narasimhan–Ramanan which identifies [Formula: see text] with [Formula: see text]. The singular points of [Formula: see text] correspond to abelian representations, and we prove that each has a neighborhood in [Formula: see text] homeomorphic to a cone on [Formula: see text].

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