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.

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.

We prove that if Y is a closed, oriented 3-manifold with first homology H_1(Y;ℤ) of order less than 5, then there is an irreducible representation π_1(Y) →SL(2,ℂ) unless Y is homeomorphic to S^3, a lens space, or ℝℙ^3 #ℝℙ^3. By previous work it suffices to consider the case H_1(Y;ℤ) ≅ℤ/4ℤ, which we accomplish using holonomy perturbation techniques in instanton Floer homology.

We prove that if an integer homology three-sphere contains an embedded incompressible torus, then its fundamental group admits irreducible SU(2)$SU(2)$-representations.

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.

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.

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 that are not hyperbolic. As an application, we give examples of hyperbolic 3-manifolds that 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.

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.

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.

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 Ozsv\'ath-Szab\'o 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 Ozsv\'ath-Szab\'o spectral sequence for an infinite class of prime knots; and to show that higher differentials of both the Kronheimer-Mrowka and the Ozsv\'ath-Szab\'o spectral sequences necessarily lower the delta grading for all pretzel knots.

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 Ozsv\'ath, Stipsicz, and Szab\'o's Upsilon invariant, allows us to determine the exact cobordism distance between torus knots with braid index two and six.

Zusammenfassung In Frankreich in aller Munde, in Deutschland noch wenig beachtet. Für junge Mathematiker bietet sie neue Berufsfelder, unser tägliches Leben wird sie verändern, viele technische Bereiche werden durch sie revolutioniert: die künstliche Intelligenz. Der Mathematiker Cédric Villani möchte dieses wichtige Gebiet in Frankreich und noch lieber in ganz Europa nach vorne bringen und diskutierte im Vorfeld der Gauß-Vorlesung in Regensburg mit Wissenschaftlern und Vertretern der Industrie. Die wichtigsten Anliegen von Villanis Initiative haben sich inzwischen zu Eckpunkten der Politik der deutschen Bundesregierung entwickelt.

Zusammenfassung Cédric Villani war der Hauptredner der Gauß-Vorlesung der DMV in Regensburg im Oktober 2017. Da Villani im Juni zuvor für die Partei En Marche! von Emmanuel Macron zum Abgeordneten des französischen Parlaments gewählt wurde, war es eine kleine Sensation, dass er den lange versprochenen Termin einhalten konnte. Obwohl die deutsche Presse die Veranstaltung nahezu ignorierte, war der wunderschöne Neuhaussaal des Regensburger Theaters fast bis auf den letzten Platz gefüllt und zur Vermeidung von Überfüllung mussten sogar kostenlose Platzkarten ausgegeben werden.

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 Ozsv\'ath, Stipsicz, and Szab\'o. 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.

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.

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.

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.

Background: Patellofemoral instability may lead to osteoarthritis, anterior knee pain, and patellar luxation. The purpose of this study was to conduct an exploratory investigation into the difference of patellar kinematics of healthy knees during extension/flexion cycles in neutral, varus and valgus alignmentMethods: The three-dimensional patellar kinematics of 10 lower extremities of whole body cadavers were examined during passive motion, in neutral position, and under valgus and varus stress. Kinematics was recorded by means of an optical computer navigation system.Results: The study samples did not significantly differ with regard to mediolateral patellar shift and epicondylar distance. Varus stress led to significantly higher external rotation than valgus stress (P = 0.04) and to a significantly higher lateral patellar tilt than neutral position (P = 0.016) and valgus stress (P = 0.016). No difference was found between valgus stress and neutral position.Conclusion: Analysis of tibiofemoral alignment alone is insufficient for predicting patellar kinematics. (C) 2017 Elsevier B.V. All rights reserved.

We prove the existence of a new algorithm for 3-sphere recognition based on Groebner basis methods applied to the variety of SL(2,)-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 SL(2,). This result, and hence our algorithm, build on the geometrisation theorem of 3-manifolds.

We associate a moduli problem to a colored trivalent graph; such graphs, when planar, appear in the state-sum description of the quantum sl(N) knot polynomial due to Murakami, Ohtsuki, and Yamada. We discuss how the resulting moduli space can be thought of a representation variety. We show that the Euler characteristic of the moduli space is equal to the quantum sl(N) polynomial of the graph evaluated at unity. Possible extensions of the result are also indicated.