arXiv subjects

Raphael Zentner

Publications and source records attributed to Raphael Zentner.

At least 19 recordsLinked to original sources

arXiv2025

Homology lens spaces and $\mathrm{SL}(2,\mathbb{C})$

We prove that if $Y$ is a closed, oriented 3-manifold with first homology $H_1(Y;\mathbb{Z})$ of order less than $5$, then there is an irreducible representation $\pi_1(Y) \to \mathrm{SL}(2,\mathbb{C})$ unless $Y$ is homeomorphic to $S^3$, a lens space, or $\mathbb{RP}^3 \# \mathbb{RP}^3$. By previous work it suffices to consider the case $H_1(Y;\mathbb{Z}) \cong \mathbb{Z}/4\mathbb{Z}$, which we accomplish using holonomy perturbation techniques in instanton Floer homology.

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arXiv2023

Rational homology 3-spheres and SL(2,$\mathbb{C}$) representations

We use instanton gauge theory to prove that if $Y$ is a closed, orientable $3$-manifold such that $H_1(Y;\mathbb{Z})$ is nontrivial and either $2$-torsion or $3$-torsion, and if $Y$ is neither $\#^r \mathbb{RP}^3$ for some $r\geq 1$ nor $\pm L(3,1)$, then there is an irreducible representation $\pi_1(Y) \to \mathrm{SL}(2,\mathbb{C})$. We apply this to show that the Kauffman bracket skein module of a non-prime 3-manifold has nontrivial torsion whenever two of the prime summands are different from $\mathbb{RP}^3$, answering a conjecture of Przytycki (Kirby problem 1.92(F)) unless every summand but one is $\mathbb{RP}^3$. As part of the proof in the $2$-torsion case, we also show that if $M$ is a compact, orientable $3$-manifold with torus boundary whose rational longitude has order 2 in $H_1(M)$, then $M$ admits a degree-1 map onto the twisted $I$-bundle over the Klein bottle.

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arXiv2022

Rational homology ribbon cobordism is a partial order

We show that ribbon rational homology cobordism is a partial order within the class of irreducible 3-manifolds. This makes essential use of the methods recently employed by Ian Agol to show that ribbon knot concordance is a partial order.

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arXiv2021

Surgery obstructions and character varieties

We provide infinitely many rational homology 3-spheres with weight-one fundamental groups which do not arise from Dehn surgery on knots in $S^3$. In contrast with previously known examples, our proofs do not require any gauge theory or Floer homology. Instead, we make use of the $SU(2)$ character variety of the fundamental group, which for these manifolds is particularly simple: they are all $SU(2)$-cyclic, meaning that every $SU(2)$ representation has cyclic image.

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arXiv2021

The Montesinos trick for proper rational tangle replacement

Recently Iltgen, Lewark and Marino introduced the concept of a proper rational tangle replacement and the corresponding notion of the proper rational unknotting number. In this note we derive a version of the Montesinos trick for proper rational tangle replacement and use it to study knots with proper rational unknotting number one. We prove that knots with proper rational unknotting number one are prime and classify the alternating knots with proper rational unknotting number one. We also study Montesinos knots with proper rational unknotting number one.

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arXiv2021

Toroidal homology spheres and SU(2)-representations

We prove that if an integer homology three-sphere contains an embedded incompressible torus, then its fundamental group admits irreducible SU(2)-representations. Our methods use instanton Floer homology, and in particular the surgery exact triangle, holonomy perturbations, and a non-vanishing result due to Kronheimer-Mrowka, as well as results about surgeries on cables due to Gordon.

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arXiv2020

A menagerie of SU(2)-cyclic 3-manifolds

We classify $SU(2)$-cyclic and $SU(2)$-abelian 3-manifolds, for which every representation of the fundamental group into $SU(2)$ has cyclic or abelian image respectively, among geometric 3-manifolds which are not hyperbolic. As an application, we give examples of hyperbolic 3-manifolds which do not admit degree-1 maps to any Seifert fibered manifold other than $S^3$ or a lens space. We also produce infinitely many one-cusped hyperbolic manifolds with at least four $SU(2)$-cyclic Dehn fillings, one more than the number of cyclic fillings allowed by the cyclic surgery theorem.

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arXiv2020

SU(2)-cyclic surgeries and the pillowcase

We study knots in $S^3$ with infinitely many $SU(2)$-cyclic surgeries, which are Dehn surgeries such that every representation of the resulting fundamental group into $SU(2)$ has cyclic image. We show that for every such nontrivial knot $K$, its set of $SU(2)$-cyclic slopes is bounded and has a unique limit point, which is both a rational number and a boundary slope for $K$. We also show that such knots are prime and have infinitely many instanton L-space surgeries. Our methods include the application of holonomy perturbation techniques to instanton knot homology, using a strengthening of recent work by the second author.

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arXiv2020

Khovanov width and dealternation number of positive braid links

We give asymptotically sharp upper bounds for the Khovanov width and the dealternation number of positive braid links, in terms of their crossing number. The same braid-theoretic technique, combined with Ozsváth, Stipsicz, and Szabó's Upsilon invariant, allows us to determine the exact cobordism distance between torus knots with braid index two and six.

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arXiv2018

Integer homology 3-spheres admit irreducible representations in SL(2,C)

We prove that the fundamental group of any integer homology 3-sphere different from the 3-sphere admits irreducible representations of its fundamental group in SL(2,C). For hyperbolic integer homology spheres this comes with the definition, and for Seifert fibered integer homology spheres this is well known. We prove that the splicing of any two non-trivial knots in the 3-sphere admits an irreducible SU(2)-representation. By work of Boileau, Rubinstein, and Wang, the general case follows. Using a result of Kuperberg, we get the corollary that the problem of 3-sphere recognition is in the complexity class coNP, provided the generalised Riemann hypothesis holds. To prove our result, we establish a topological fact about the image of the SU(2)-representation variety of a non-trivial knot complement into the representation variety of its boundary torus, a pillowcase. For this, we use holonomy perturbations of the Chern-Simons function in an exhaustive way - we show that any area-preserving self-map of the pillowcase fixing the four singular points, and which is isotopic to the identity, can be C^0-approximated by maps which are realised geometrically through holonomy perturbations of the flatness equation in a thickened torus. To conclude, we use a stretching argument in instanton gauge theory, and a non-vanishing result of Kronheimer and Mrowka for Donaldson's invariants of a 4-manifold which contains the 0-surgery of a knot as a splitting hypersurface.

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arXiv2017

Integrable almost complex structures in principal bundles and holomorphic curves

We consider principal fibre bundles with a given connection and construct almost complex structures on the total space if the adjoint bundle is isomorphic to the tangent bundle of the base. We derive the integrability condition. If the structure group is compact, then a choice of an ad-invariant inner product on its Lie algebra gives naturally the structure of a Riemannian manifold to the base. The integrability condition is then expressed in geometric terms. In particular we get a relation to hyperbolic geometry if the structure group is SU(2) or SO(3). The bundle of orthonormal frames of a hyperbolic oriented 3-manifold is naturally a complex manifold. If the base is geodesically complete and connected, then we can endow the total space with a locally free transitive holomorphic action of the complexified structure group. We then get some restrictions for holomorphic maps from Riemann surfaces to the total space. If the pull-back of a canonical Lie algebra valued 1-form on the total space is of scalar form, then the holomorphic map factorises through an elliptic curve. If the induced map to the base is conformal, then the associated holomorphic map with value in $P^2 \mathbb{C}$ factorises through a smooth quadric.

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arXiv2017

A class of knots with simple $SU(2)$ representations

We call a knot in the 3-sphere $SU(2)$-simple if all representations of the fundamental group of its complement which map a meridian to a trace-free element in $SU(2)$ are binary dihedral. This is a generalisation of being a 2-bridge knot. Pretzel knots with bridge number $\geq 3$ are not $SU(2)$-simple. We provide an infinite family of knots $K$ with bridge number $\geq 3$ which are $SU(2)$-simple. One expects the instanton knot Floer homology $I^\natural(K)$ of a $SU(2)$-simple knot to be as small as it can be -- of rank equal to the knot determinant $\det(K)$. In fact, the complex underlying $I^\natural(K)$ is of rank equal to $\det(K)$, provided a genericity assumption holds that is reasonable to expect. Thus formally there is a resemblance to strong L-spaces in Heegaard Floer homology. For the class of $SU(2)$-simple knots that we introduce this formal resemblance is reflected topologically: The branched double covers of these knots are strong L-spaces. In fact, somewhat surprisingly, these knots are alternating. However, the Conway spheres are hidden in any alternating diagram. With the methods we use, we show that an integer homology 3-sphere which is a graph manifold always admits irreducible representations of its fundamental group.

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arXiv2017

A vanishing result for a Casson-type instanton invariant

Casson-type invariants emerging from Donaldson theory over certain negative definite 4-manifolds were recently suggested by Andrei Teleman. These are defined by a count of a zero-dimensional moduli space of flat instantons. Motivated by the cobordism program of proving Witten's conjecture, we use a moduli space of PU(2) Seiberg-Witten monopoles to exhibit an oriented one-dimensional cobordism of the instanton moduli space to the empty space. The Casson-type invariant must therefore vanish.

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arXiv2016

A new algorithm for 3-sphere recognition

We prove the existence of a new algorithm for 3-sphere recognition based on Groebner basis methods applied to the variety of $\text{\em SL}(2,\C)$-representation of the fundamental group. An essential input is a recent result of the second author, stating that any integer homology 3-sphere different from the 3-sphere admits an irreducible representation of its fundamental group in $\text{\em SL}(2,\C)$. This result, and hence our algorithm, build on the geometrisation theorem of 3-manifolds.

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arXiv2015

Knot concordances and alternating knots

There is an infinitely generated free subgroup of the smooth knot concordance group with the property that no nontrivial element in this subgroup can be represented by an alternating knot. This subgroup has the further property that every element is represented by a topologically slice knot.

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arXiv2015

Alternating numbers of torus knots with small braid index

We calculate the alternating number of torus knots with braid index 4 and less. For the lower bound, we use the upsilon-invariant recently introduced by Ozsváth, Stipsicz, and Szabó. For the upper bound, we use a known bound for braid index $3$ and a new bound for braid index $4$. Both bounds coincide, so that we obtain a sharp result.

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arXiv2013

On spectral sequences from Khovanov homology

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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arXiv2012

Representation spaces of pretzel knots

We study the representation spaces $R(K;\bf{i})$ as appearing in Kronheimer and Mrowka's framed instanton knot Floer homology, for a class of pretzel knots. In particular, for pretzel knots $P(p,q,r)$ with $p, q, r$ pairwise coprime, these appear to be non-degenerate and comprise representations in SU(2) that are not binary dihedral.

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