Traceless SU(2) representations of 2-stranded tangles

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.

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Hodge polynomials of the SL(2, ℂ)-character variety of an elliptic curve with two marked points

International Journal of Mathematics ◽

10.1142/s0129167x14501250 ◽

2014 ◽

Vol 25 (14) ◽

pp. 1450125 ◽

Cited By ~ 3

Author(s):

Marina Logares ◽

Vicente Muñoz

Keyword(s):

Elliptic Curve ◽

Moduli Space ◽

Betti Numbers ◽

Conjugacy Classes ◽

Character Variety ◽

Higgs Bundles ◽

Hodge Numbers ◽

Hodge Polynomials ◽

Marked Points

We compute the Hodge polynomials for the moduli space of representations of an elliptic curve with two marked points into SL(2, ℂ). When we fix the conjugacy classes of the representations around the marked points to be diagonal and of modulus one, the character variety is diffeomorphic to the moduli space of strongly parabolic Higgs bundles, whose Betti numbers are known. In that case we can recover some of the Hodge numbers of the character variety. We extend this result to the moduli space of doubly periodic instantons.

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Character varieties of even classical pretzel knots

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/012.2019.56.4.1439 ◽

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.

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Character varieties of odd classical pretzel knots

International Journal of Mathematics ◽

10.1142/s0129167x1850060x ◽

2018 ◽

Vol 29 (09) ◽

pp. 1850060 ◽

Cited By ~ 1

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.

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An Explicit View of the Hitchin Fibration on the Betti Side for ℙ1 minus Five Points

Geometry and Physics: Volume II ◽

10.1093/oso/9780198802020.003.0029 ◽

2018 ◽

pp. 705-724

Author(s):

Carlos T. Simpson

Keyword(s):

Conjugacy Classes ◽

Character Variety ◽

Tubular Neighbourhood ◽

Local Systems ◽

Hitchin Fibration ◽

Dual Complex

The dual complex of the divisor at infinity of the character variety of local systems on P1−{t1,…,t5} with monodromies in prescribed conjugacy classes Ci⊂SL2(C), was shown by Komyo to be the sphere S3. This chapter compares in some detail the projection from a tubular neighbourhood to this dual complex, with the corresponding Hitchin fibration at infinity.

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Nimble evolution for pretzel Khovanov polynomials

The European Physical Journal C ◽

10.1140/epjc/s10052-019-7303-5 ◽

2019 ◽

Vol 79 (10) ◽

Cited By ~ 2

Author(s):

Aleksandra Anokhina ◽

Alexei Morozov ◽

Aleksandr Popolitov

Keyword(s):

Frequency Doubling ◽

Complex Variables ◽

Point Of View ◽

Laurent Polynomials ◽

Frequency Multiplication ◽

Linear Dynamics ◽

Pretzel Knots ◽

Pure Phase ◽

Non Linear ◽

The One

Abstract We conjecture explicit evolution formulas for Khovanov polynomials, which for any particular knot are Laurent polynomials of complex variables q and T, for pretzel knots of genus g in some regions in the space of winding parameters $$n_0, \dots , n_g$$n0,⋯,ng. Our description is exhaustive for genera 1 and 2. As previously observed Anokhina and Morozov (2018), Dunin-Barkowski et al. (2019), evolution at $$T\ne -1$$T≠-1 is not fully smooth: it switches abruptly at the boundaries between different regions. We reveal that this happens also at the boundary between thin and thick knots, moreover, the thick-knot domain is further stratified. For thin knots the two eigenvalues 1 and $$\lambda = q^2 T$$λ=q2T, governing the evolution, are the standard T-deformation of the eigenvalues of the R-matrix 1 and $$-q^2$$-q2. However, in thick knots’ regions extra eigenvalues emerge, and they are powers of the “naive” $$\lambda $$λ, namely, they are equal to $$\lambda ^2, \dots , \lambda ^g$$λ2,⋯,λg. From point of view of frequencies, i.e. logarithms of eigenvalues, this is frequency doubling (more precisely, frequency multiplication) – a phenomenon typical for non-linear dynamics. Hence, our observation can signal a hidden non-linearity of superpolynomial evolution. To give this newly observed evolution a short name, note that when $$\lambda $$λ is pure phase the contributions of $$\lambda ^2, \dots , \lambda ^g$$λ2,⋯,λg oscillate “faster” than the one of $$\lambda $$λ. Hence, we call this type of evolution “nimble”.

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B. The Non-Linear Calculations for Pulsating Stars

Symposium - International Astronomical Union ◽

10.1017/s0074180900019008 ◽

1967 ◽

Vol 28 ◽

pp. 105-176

Author(s):

Robert F. Christy

Keyword(s):

Major Part ◽

Afternoon Session ◽

Discussion Session ◽

Pulsating Stars ◽

Fourth Symposium ◽

Non Linear ◽

Considerable Discussion ◽

Informal Approach ◽

Preceding Session

(Ed. note: The custom in these Symposia has been to have a summary-introductory presentation which lasts about 1 to 1.5 hours, during which discussion from the floor is minor and usually directed at technical clarification. The remainder of the session is then devoted to discussion of the whole subject, oriented around the summary-introduction. The preceding session, I-A, at Nice, followed this pattern. Christy suggested that we might experiment in his presentation with a much more informal approach, allowing considerable discussion of the points raised in the summary-introduction during its presentation, with perhaps the entire morning spent in this way, reserving the afternoon session for discussion only. At Varenna, in the Fourth Symposium, several of the summaryintroductory papers presented from the astronomical viewpoint had been so full of concepts unfamiliar to a number of the aerodynamicists-physicists present, that a major part of the following discussion session had been devoted to simply clarifying concepts and then repeating a considerable amount of what had been summarized. So, always looking for alternatives which help to increase the understanding between the different disciplines by introducing clarification of concept as expeditiously as possible, we tried Christy's suggestion. Thus you will find the pattern of the following different from that in session I-A. I am much indebted to Christy for extensive collaboration in editing the resulting combined presentation and discussion. As always, however, I have taken upon myself the responsibility for the final editing, and so all shortcomings are on my head.)

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