Knot polynomials and knot homologies

By Jacob Rasmussen. 2005

115 citations recorded by the source corpus.

This is an expository paper discussing some parallels between the Khovanov and knot Floer homologies. We describe the formal similarities between the theories and give some examples which illustrate a somewhat mysterious correspondence between them.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c0f0e6ca6c6c417f9c7bfe2c40200249b7816209

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In Computational Theory and Mathematics

In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

Grid Homology for Knots and Links (2015, background): “There is a conjectured spectral sequence from reduced Khovanov homology to ĤFK, see [193]; and from reduced HOMFLY homology to ĤFK, see [36].”
On the geography and botany of knot Floer homology (2014, background): “There are a pair of long exact sequences for this theory described by Rasmussen [81], though our conventions are consistent with Manolescu and Ozsváth [52, Proposition 2.”
A remark on rasmussen's invariant of knots (2005, background): “Much of this is explained in the survey paper [6].”
On the Khovanov and knot Floer homologies of quasi-alternating links (2007, background): “We call such links Khovanov homologically thin (over R), or Floer homologically thin (over R), depending on which theory we refer to. Various versions of these definitions appeared in [2], [27], [6], [ 28 ].”
CALCULATING BAR-NATAN'S CHARACTERISTIC TWO KHOVANOV HOMOLOGY (2004, background): “Using results of Shumakovitch [8] (see also [7]) in the mod 2 c ase we have KH F2 (L) ∼= K̃H ∗,∗ F2 (L)⊗A from which it follows that KhF2(L) = K̃hF2(L)(q + q ) = (q + q)q(1 + (1 + tq(2)) ∑”
Heegaard Floer homology and genus one, one‐boundary component open books (2008, background): “The next proposition follows immediately from the definition of Khovanov homology, and can be found in the form below in [24, 42].”
Fivebranes and 3-manifold homology (2016, background): “Both of these two theories are extremely concrete and computation-friendly, which immediately led to a number of surprising observations [4, 5].”
Metabelian representations, twisted Alexander polynomials, knot slicing, and mutation (2008, unknown): “A few more recent articles include [4,5,9,32,34,35].”
Knot homology groups from instantons (2008, background): “Rasmussen [39] has observed that the reduced Khovanov homology coincides with Heegaard-Floer knot homology group ĤFK (K), defined by Ozsváth and Szabó in [33] and by Rasmussen in [38], for many knots, but not for the (4, 5) torus knot.”
Computations of Heegaard-Floer knot homology (2006, unknown): “and Rasmussen conjectures the following [25]:”
Odd Khovanov homology (2007, unknown): “The maps i , and ı are all homogenous with respect to the bigrading; for a precise statement of the grading shifts, see eg Manolescu and the first author’s [10] or the second author’s [14].”
AN INTRODUCTION TO KNOT FLOER HOMOLOGY (2014, unknown): “• There are structural parallels to Khovanov’s sl (2) homology, discussed in Rasmussen’s survey [Ras05].”
Homological thickness and stability of torus knots (2005, background): “g applications in low-dimensional topology. For instance, the short proof of the Milnor conjecture by Rasmussen in [19], as well as the proof of the existence of exotic differential structures on R4 ([20]), which were previously accessible only by gauge theory. The advantage of Khovanov homology the”
The Ozsváth-Szabó and Rasmussen Concordance Invariants are not Equal (2005, background): “) for all knots, K. In light of the above list, the formal properties that the two invariants share, and several other striking connections between Khovanov’s homology theory and Ozsv´athSzabo theory [33, 26, 16, 31], there was justiﬁed hope that the above conjecture could be true. However, we will ”