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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenThu, 30 Nov 2023 20:13:20 +0000Berge–Gabai knots and L–space satellite operations
https://resolver.caltech.edu/CaltechAUTHORS:20150327-060832382
Authors: Hom, Jennifer; Lidman, Tye; Vafaee, Faramarz
Year: 2014
DOI: 10.2140/agt.2014.14.3745
Let P(K) be a satellite knot where the pattern P is a Berge–Gabai knot (ie a knot in the solid torus with a nontrivial solid torus Dehn surgery) and the companion K is a nontrivial knot in S^3. We prove that P(K) is an L–space knot if and only if K is an L–space knot and P is sufficiently positively twisted relative to the genus of K. This generalizes the result for cables due to Hedden [Int. Math. Res. Not. 2009 (2009) 2248–2274] and Hom [Algebr. Geom. Topol. 11 (2011) 219–223].https://authors.library.caltech.edu/records/km8rp-9s541Seifert surfaces distinguished by sutured Floer homology but not its Euler characteristic
https://resolver.caltech.edu/CaltechAUTHORS:20150416-101226466
Authors: Vafaee, Faramarz
Year: 2015
DOI: 10.1016/j.topol.2015.01.005
In this paper we find a family of knots with trivial Alexander polynomial, and construct two non-isotopic Seifert surfaces for each member in our family. In order to distinguish the surfaces we study the sutured Floer homology invariants of the sutured manifolds obtained by cutting the knot complements along the Seifert surfaces. Our examples provide the first use of sutured Floer homology, and not merely its Euler characteristic (a classical torsion), to distinguish Seifert surfaces. Our technique uses a version of Floer homology, called "longitude Floer homology" in a way that enables us to bypass the computations related to the SFH of the complement of a Seifert surface.https://authors.library.caltech.edu/records/5wygv-60197A slicing obstruction from the 10/8 theorem
https://resolver.caltech.edu/CaltechAUTHORS:20161117-085125275
Authors: Donald, Andrew; Vafaee, Faramarz
Year: 2016
DOI: 10.1090/proc/13056
From Furuta's 10/8 theorem, we derive a smooth slicing obstruction for knots in S^3 using a spin 4-manifold whose boundary is 0-surgery on a knot. We show that this obstruction is able to detect torsion elements in the smooth concordance group and find topologically slice knots which are not smoothly slice.https://authors.library.caltech.edu/records/hn70w-fjq78(1,1) L-space knots
https://resolver.caltech.edu/CaltechAUTHORS:20180516-100045096
Authors: Greene, Joshua Evan; Lewallen, Sam; Vafaee, Faramarz
Year: 2018
DOI: 10.1112/S0010437X17007989
We characterize the (1,1) knots in the 3-sphere and lens spaces that admit non-trivial L-space surgeries. As a corollary, 1-bridge braids in these manifolds admit non-trivial L-space surgeries. We also recover a characterization of the Berge manifold among 1-bridge braid exteriors.https://authors.library.caltech.edu/records/yce57-ktn67Null surgery on knots in L-spaces
https://resolver.caltech.edu/CaltechAUTHORS:20180308-071823523
Authors: Ni, Yi; Vafaee, Faramarz
Year: 2019
DOI: 10.1090/tran/7510
Let K be a knot in an L-space Y with a Dehn surgery to a surface bundle over S¹. We prove that K is rationally fibered, that is, the knot complement admits a fibration over S¹. As part of the proof, we show that if K C Y has a Dehn surgery to S¹ x S², then K is rationally fibered. In the case that K admits some S¹ x S² surgery, K is Floer simple, that is, the rank of HFK(Y,K) is equal to the order of H₁(Y). By combining the latter two facts, we deduce that the induced contact structure on the ambient manifold Y is tight. In a different direction, we show that if K is a knot in an L-space Y, then any Thurston norm minimizing rational Seifert surface for K extends to a Thurston norm minimizing surface in the manifold obtained by the null surgery on K (i.e., the unique surgery on K with b₁ > 0).https://authors.library.caltech.edu/records/p2ny4-2hm61The prism manifold realization problem
https://resolver.caltech.edu/CaltechAUTHORS:20180308-070142351
Authors: Ballinger, William; Hsu, Chloe Ching-Yun; Mackey, Wyatt; Ni, Yi; Ochse, Tynan; Vafaee, Faramarz
Year: 2020
DOI: 10.2140/agt.2020.20.757
The spherical manifold realization problem asks which spherical three-manifolds arise from surgeries on knots in S³. In recent years, the realization problem for C–, T–, O– and I–type spherical manifolds has been solved, leaving the D–type manifolds (also known as the prism manifolds) as the only remaining case. Every prism manifold can be parametrized as P(p,q) for a pair of relatively prime integers p>1 and q. We determine a list of prism manifolds P(p,q) that can possibly be realized by positive integral surgeries on knots in S³ when q<0. Based on the forthcoming work of Berge and Kang, we are confident that this list is complete. The methodology undertaken to obtain the classification is similar to that of Greene for lens spaces.https://authors.library.caltech.edu/records/g8t8s-r6903