Representation spaces of pretzel knots

Zentner, Raphael

doi:10.2140/agt.2011.11.2941

Cited by 6 publications

(9 citation statements)

References 22 publications

(52 reference statements)

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“…The space R(S 3 , K) for K any pretzel knot is identified by Raphael Zentner in [21]. We sketch what happens when one decomposes a pretzel knot along a Conway sphere, to obtain two tangles.…”

Section: Tangles In Pretzel Knot Complementsmentioning

confidence: 99%

Traceless SU(2) representations of 2-stranded tangles

Fukumoto¹,

Kirk²,

Pinzón-Caicedo³

2016

Math. Proc. Camb. Phil. Soc.

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Given a 2-stranded tangle in a $\ZZ/2$ homology ball, $T\subset Y$, we investigate the character variety $R(Y,T)$ of conjugacy classes of traceless SU(2) representations of $\pi_1(Y\setminus T)$. 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 S^3. 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 {\it 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.Comment: 31 pages, color figure

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Section: Tangles In Pretzel Knot Complementsmentioning

confidence: 99%

Traceless SU(2) representations of 2-stranded tangles

Fukumoto¹,

Kirk²,

Pinzón-Caicedo³

2016

Math. Proc. Camb. Phil. Soc.

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“…It is conjectured that there is similar spectral sequence in that context.) Zentner [37] showed that for some alternating pretzel knots there are non-binary dihedral traceless representations (in contrast to 2-bridge knots), so that for these families one expects there to be non-trivial differentials in the singular instanton chain complex.…”

Section: Introductionmentioning

confidence: 99%

The pillowcase and perturbations of traceless representations of knot groups

2014

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Abstract. We introduce explicit holonomy perturbations of the Chern-Simons functional on a 3-ball containing a pair of unknotted arcs. These perturbations give us a concrete local method for making the moduli spaces of flat singular SO(3) connections relevant to Kronheimer and Mrowka's singular instanton knot homology non-degenerate. The mechanism for this study is a (Lagrangian) intersection diagram which arises, through restriction of representations, from a tangle decomposition of a knot. When one of the tangles is trivial, our perturbations allow us to study isolated intersections of two Lagrangians to produce minimal generating sets for singular instanton knot homology. The (symplectic) manifold where this intersection occurs corresponds to the traceless character variety of the four-punctured 2-sphere, which we identify with the familiar pillowcase. We investigate the image in this pillowcase of the traceless representations of tangles obtained by removing a trivial tangle from 2-bridge knots and torus knots. Using this, we compute the singular instanton homology of a variety of torus knots. In many cases, our computations allow us to understand nontrivial differentials in the spectral sequence from Khovanov homology to singular instanton homology.

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“…We list a few cases explicitly. The claims on the representation spaces of pretzel knots can be found in [3] and [19].…”

Section: 5mentioning

confidence: 99%

“…• For the pretzel knot P (−3, 5, 7) we have rk(Khr(P (−3, 5, 7))) = 15, whereas R(P (−3, 5, 7); i) contains the conjugacy class of the reducible and 16 conjugacy classes of irreducible non binary dihedral representations (see the table of the example in [19] where 3 errors occur that yield a total error of 1 which multiplied by two gave the wrong claim of 18 conjugacy classes).…”

Section: 5mentioning

confidence: 99%

On spectral sequences from Khovanov homology

Lobb

Zentner

2020

Algebr. Geom. Topol.

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There are a number of homological knot invariants, each satisfying an unoriented skein exact sequence, which can be realized as the limit page of a spectral sequence starting at a version of the Khovanov chain complex. Compositions of elementary 1-handle movie moves induce a morphism of spectral sequences. These morphisms remain unexploited in the literature, perhaps because there is still an open question concerning the naturality of maps induced by general movies.In this paper we focus on the spectral sequences due to Kronheimer-Mrowka from Khovanov homology to instanton knot Floer homology, and on that due to Ozsváth-Szabó to the Heegaard-Floer homology of the branched double cover. For example, we use the 1-handle morphisms to give new information about the filtrations on the instanton knot Floer homology of the (4, 5)-torus knot, determining these up to an ambiguity in a pair of degrees; to determine the Ozsváth-Szabó spectral sequence for an infinite class of prime knots; and to show that higher differentials of both the Kronheimer-Mrowka and the Ozsváth-Szabó spectral sequences necessarily lower the delta grading for all pretzel knots.

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Representation spaces of pretzel knots

Cited by 6 publications

References 22 publications

Traceless SU(2) representations of 2-stranded tangles

Traceless SU(2) representations of 2-stranded tangles

The pillowcase and perturbations of traceless representations of knot groups

On spectral sequences from Khovanov homology

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