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Register a new userL-space fillings and generalized solid tori
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Thomas J. Gillespie
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Much work has been done recently towards trying to understand the topological significance of being an L-space. Building on work of Rasmussen and Rasmussen, we give a topological characterisation of Floer simple manifolds such that all non-longitudinal fillings are L-spaces. We use this to partially classify L-space twisted torus knots in $S^1 times S^2$ and resolve a question asked by Rasmussen and Rasmussen.
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We prove that instanton L-space knots are fibered and strongly quasipositive. Our proof differs conceptually from proofs of the analogous result in Heegaard Floer homology, and includes a new decomposition theorem for cobordism maps in framed instanton Floer homology akin to the $textrm{Spin}^c$ decompositions of cobordism maps in other Floer homology theories. As our main application, we prove (modulo a mild nondegeneracy condition) that for $r$ a positive rational number and $K$ a nontrivial knot in the $3$-sphere, there exists an irreducible homomorphism [pi_1(S^3_r(K)) to SU(2)] unless $r geq 2g(K)-1$ and $K$ is both fibered and strongly quasipositive, broadly generalizing results of Kronheimer and Mrowka. We also answer a question of theirs from 2004, proving that there is always an irreducible homomorphism from the fundamental group of 4-surgery on a nontrivial knot to $SU(2)$. In another application, we show that a slight enhancement of the A-polynomial detects infinitely many torus knots, including the trefoil.
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