New proofs of certain finite filling results via Khovanov homology

We give a Khovanov homology proof that hyperbolic twist knots do not admit non-trivial Dehn surgeries with finite fundamental group.Comment: 21 pages, 18 figures. Version 2: Revised and expanded per referee's comments (including a new title). This version to appear in Quantum Topolog

On cabled knots, Dehn surgery, and left-orderable fundamental groups

Previous work of the authors establishes a criterion on the fundamental group of a knot complement that determines when Dehn surgery on the knot will have a fundamental group that is not left-orderable. We provide a refinement of this criterion by introducing the notion of a decayed knot; it is shown that Dehn surgery on decayed knots produces surgery manifolds that have non-left-orderable fundamental group for all sufficiently positive surgeries. As an application, we prove that sufficiently positive cables of decayed knots are always decayed knots. These results mirror properties of L-space surgeries in the context of Heegaard Floer homology.Comment: 11 page

Knots with identical Khovanov homology

We give a recipe for constructing families of distinct knots that have identical Khovanov homology and give examples of pairs of prime knots, as well as infinite families, with this property.Comment: v2: major revision. v3: typos corrected, minor clarifications. 12 pages, many figure

A calculus for bordered Floer homology

We consider a class of manifolds with torus boundary admitting bordered Heegaard Floer homology of a particularly simple form, namely, the type D structure may be described graphically by a disjoint union of loops. We develop a calculus for studying bordered invariants of this form and, in particular, provide a complete description of slopes giving rise to L-space Dehn fillings as well as necessary and sufficient conditions for L-spaces resulting from identifying two such manifolds along their boundaries. As an application, we show that Seifert fibered spaces with torus boundary fall into this class, leading to a proof that, among graph manifolds containing a single JSJ torus, the property of being an L-space is equivalent to non-left-orderability of the fundamental group and to the non-existence of a coorientable taut foliation.Comment: 79 pages, 14 figures, uses tik

On the geography and botany of knot Floer homology

This note explores two questions: (1) Which bigraded groups arise as the knot Floer homology of a knot in the three-sphere? (2) Given a knot, how many distinct knots share its Floer homology? Regarding the first, we show there exist bigraded groups satisfying all previously known constraints of knot Floer homology which do not arise as the invariant of a knot. This leads to a new constraint for knots admitting lens space surgeries, as well as a proof that the rank of knot Floer homology detects the trefoil knot. For the second, we show that any non-trivial band sum of two unknots gives rise to an infinite family of distinct knots with isomorphic knot Floer homology. We also prove that the fibered knot with identity monodromy is strongly detected by its knot Floer homology, implying that Floer homology solves the word problem for mapping class groups of surfaces with non-empty boundary. Finally, we survey some conjectures and questions and, based on the results described above, formulate some new ones.Comment: 39 pages; 6 figures. Version 2: references added, minor changes to last section. Version 3: minor edits and updates. This version accepted for publication in Selecta Mathematic

Turaev torsion, definite 4-manifolds, and quasi-alternating knots

We construct an infinite family of hyperbolic, homologically thin knots that are not quasi-alternating. To establish the latter, we argue that the branched double-cover of each knot in the family does not bound a negative definite 4-manifold with trivial first homology and bounded second betti number. This fact depends in turn on information from the correction terms in Heegaard Floer homology, which we establish by way of a relationship to, and calculation of, the Turaev torsion.Comment: 11 pages, 3 figure

Non-fibered L-space knots

We construct an infinite family of knots in rational homology spheres with irreducible, non-fibered complements, for which every non-longitudinal filling is an L-space.Comment: 4 pages, 2 figures. Version 2: expanded discussion per comments from the referee; this version to appear in Pacific Journal of Mathematic