On the geography and botany of knot Floer homology

By M. Hedden, Liam Watson. Selecta Mathematica, 2014

91 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics, Neurology

View Paper: https://www.semanticscholar.org/paper/34e9d5dab4bb47b661f9b997153d3d346225f409

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. John A. Baldwin
2. Steven Sivek
3. J. Greene

In Mathematical Physics

1. John A. Baldwin
2. Steven Sivek

In Neurology

1. John A. Baldwin
2. Steven Sivek
3. J. Greene

Citations in the corpus

• Khovanov homology detects the trefoils (2018, background): “10 can be used to give new proofs that the dimension of ĤFK detects the trefoil [HW14] and that L-space knots are prime [Krc15].”