Liam Watson

ON THE GEOGRAPHY AND BOTANY OF KNOT FLOER HOMOLOGY

This paper 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.

Graph manifolds, left-orderability and amalgamation

We show that every irreducible toroidal integer homology sphere graph manifold has a left-orderable fundamental group. This is established by way of a specialization of a result due to Bludov and Glass [Proc. Lond. Math. Soc. 99 (2009) 585–608] for the amalgamated products that arise, and in this setting work of Boyer, Rolfsen and Wiest [Ann. Inst. Fourier (Grenoble) 55 (2005) 243–288] may be applied. Our result then depends on known relations between the topology of Seifert fibred spaces and the orderability of their fundamental groups.

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.

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.

ANY TANGLE EXTENDS TO NON-MUTANT KNOTS WITH THE SAME JONES POLYNOMIAL

We show that an arbitrary tangle T can be extended to produce diagrams of two distinct knots that cannot be distinguished by the Jones polynomial. When T is a prime tangle, the resulting knots are prime. It is also shown that, in either case, the resulting pair are not mutants.

Genus one open books with non-left-orderable fundamental group

Let Y be a closed, connected, orientable three-manifold admitting a genus one open book decomposition with one boundary component. We prove that if Y is an L-space, then the fundamental group of Y is not left-orderable. This answers a question posed by John Baldwin.

Nonfibered L-space knots

We construct an infinite family of knots in rational homology spheres with irreducible, nonfibered complements, for which every nonlongitudinal filling is an L-space.