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Register a new userOn definite lattices bounded by integer surgeries along knots with slice genus at most 2
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We classify the positive definite intersection forms that arise from smooth 4-manifolds with torsion-free homology bounded by positive integer surgeries on the right-handed trefoil. A similar, slightly less complete classification is given for the (2,5)-torus knot, and analogous results are obtained for integer surgeries on knots of slice genus at most two. The proofs use input from Yang--Mills instanton gauge theory and Heegaard Floer correction terms.
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A surgery on a knot in 3-sphere is called SU(2)-cyclic if it gives a manifold whose fundamental group has no non-cyclic SU(2) representations. Using holonomy perturbations on the Chern-Simons functional, we prove that the distance of two SU(2)-cyclic surgery coefficients is bounded by the sum of the absolute values of their numerators. This is an analog of Culler-Gordon-Luecke-Shalens cyclic surgery theorem.
Baker showed that 10 of the 12 classes of Berge knots are obtained by surgery on the minimally twisted 5-chain link. In this article we enumerate all hyperbolic knots in S^3 obtained by surgery on the minimally twisted 5 chain link that realize the maximal known distances between slopes corresponding to exceptional (lens, lens), (lens, toroidal), (lens, Seifert fibred spaces) pairs. In light of Bakers work, the classification in this paper conjecturally accounts for most hyperbolic knots in S^3 realizing the maximal distance between these exceptional pairs. All examples obtained in our classification are realized by filling the magic manifold. The classification highlights additional examples not mentioned in Martelli and Petronios survey of the exceptional fillings on the magic manifold. Of particular interest, is an example of a knot with two lens space surgeries that is not obtained by filling the Berge manifold.
Using instanton Floer theory, extending methods due to Froyshov, we determine the definite lattices that arise from smooth 4-manifolds bounded by certain homology 3-spheres. For example, we show that for +1 surgery on the (2,5) torus knot, the only non-diagonal lattices that can occur are E8 and the indecomposable unimodular definite lattice of rank 12, up to diagonal summands. We require that our 4-manifolds have no 2-torsion in their homology.
We prove that the topological locally flat slice genus of large torus knots takes up less than three quarters of the ordinary genus. As an application, we derive the best possible linear estimate of the topological slice genus for torus knots with non-maximal signature invariant.
A knot in $S^3$ is rationally slice if it bounds a disk in a rational homology ball. We give an infinite family of rationally slice knots that are linearly independent in the knot concordance group. In particular, our examples are all infinite order. All previously known examples of rationally slice knots were order two.
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