Khovanov homology detects the figure-eight knot
Using Dowlin's spectral sequence from Khovanov homology to knot Floer homology, we prove that reduced Khovanov homology (over $\mathbb{Q}$) detects the figure-eight knot.
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arXiv subjects
Nathan Dowlin
Publications and source records attributed to Nathan Dowlin.
Relating tangle invariants for Khovanov homology and knot Floer homology
Ozsvath and Szabo recently constructed an algebraically defined invariant of tangles which takes the form of a DA bimodule. This invariant is expected to compute knot Floer homology. The authors have a similar construction for open braids and their plat closures which can be viewed as a filtered DA bimodule over the same algebras. For a closed diagram, this invariant computes the Khovanov homology of the knot or link. We show that forgetting the filtration, our DA bimodules are homotopy equivalent to a suitable version of the Ozsvath- Szabo bimodules. In addition to giving a relationship between tangle invariants for Khovanov homology and knot Floer homology, this gives an oriented skein exact triangle for the Ozsvath-Szabo bimodules which can be iterated to give an oriented cube of resolutions for the global construction.
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A spectral sequence from Khovanov homology to knot Floer homology
A well-known conjecture of Rasmussen states that for any knot $K$ in $S^{3}$, the rank of the reduced Khovanov homology of $K$ is greater than or equal to the rank of the reduced knot Floer homology of $K$. This rank inequality is supposed to arise as the result of a spectral sequence from Khovanov homology to knot Floer homology. Using an oriented cube of resolutions construction for a homology theory related to knot Floer homology, we prove this conjecture.
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A link invariant related to Khovanov homology and knot Floer homology
In this paper we introduce a chain complex $C_{1 \pm 1}(D)$ where D is a plat braid diagram for a knot K. This complex is inspired by knot Floer homology, but it the construction is purely algebraic. It is constructed as an oriented cube of resolutions with differential d=d_0+d_1. We show that the E_2 page of the associated spectral sequence is isomorphic to the Khovanov homology of K, and that the total homology is a link invariant which we conjecture is isomorphic to δ-graded knot Floer homology. The complex can be refined to a tangle invariant for braids on 2n strands, where the associated invariant is a bimodule over an algebra A_n. We show that A_n is isomorphic to B'(2n+1, n), the algebra used for the DA-bimodule constructed by Ozsvath and Szabo in their algebraic construction of knot Floer homology.
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A family of $\mathfrak{sl}_{n}$-like invariants in knot Floer homology
We define and study a family of link invariants $\mathit{HFK}_{n}(L)$. Although these homology theories are defined using holomorphic disc counts, they share many properties with $sl_{n}$ homology. Using these theories, we give a framework that generalizes the conjectured spectral sequence from Khovanov homology to $δ$-graded knot Floer homology. In particular, we conjecture that for all links $L$ in $S^3$ and all $n\ge 1$, there is a spectral sequence from the $sl_{n}$ homology of $L$ to $\mathit{HFK}_{n}(L)$.
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The Lee Spectral Sequence, Unknotting Number, and the Knight Move Conjecture
We show that the page at which the Lee spectral sequence collapses gives a bound on the unknotting number, u(K). In particular, for knots with u(K)<3, we show that the Lee spectral sequence must collapse at the E_2 page. An immediate corollary is that the Knight Move Conjecture is true when u(K)<3.
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A Categorification of the HOMFLY-PT Polynomial with a Spectral Sequence to Knot Floer Homology
Let $E_{k}^{F}(D)$ be the spectral sequence induced by the oriented cube of resolutions on knot Floer homology. We prove that $E_{2}^{F}(D)$ is a triply graded link invariant whose graded Euler characteristic is the HOMFLY-PT polynomial and that the higher pages are link invariants. By construction, the spectral sequence converges to knot Floer homology. We show that the rank of the torsion-free part of $E_{2}^{F}(D)$ is the rank of HOMFLY-PT homology.
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The Knot Floer Cube of Resolutions and the Composition Product
We examine the relationship between the (untwisted) knot Floer cube of resolutions and HOMFLY-PT homology. By using a filtration induced by additional basepoints on the Heegaard diagram for a knot $K$, we see that the filtered complex decomposes as a direct sum of HOMFLY-PT homologies of various subdiagrams. Jaeger's composition product formula shows that the graded Euler characteristic of this direct sum is the HOMFLY-PT polynomial of $K$.
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Knot Floer Homology and Khovanov-Rozansky Homology for Singular Links
The (untwisted) oriented cube of resolutions for knot Floer homology assigns a complex $C_{F}(S)$ to a singular resolution $S$ of a knot $K$. Manolescu conjectured that when $S$ is in braid position, the homology $H_{*}(C_{F}(S))$ is isomorphic to the HOMFLY-PT homology of $S$. Together with a naturality condition on the induced edge maps, this conjecture would prove the spectral sequence from HOMFLY-PT homology to knot Floer homology. Using a basepoint filtration on $C_{F}(S)$, a recursion formula for HOMFLY-PT homology, and additional $sl_{n}$-like differentials on $C_{F}(S)$, we prove this conjecture.
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The Role of Mergers in Early-type Galaxy Evolution and Black Hole Growth
Models of galaxy formation invoke the major merger of gas-rich progenitor galaxies as the trigger for significant phases of black hole growth and the associated feedback that suppresses star formation to create red spheroidal remnants. However, the observational evidence for the connection between mergers and active galactic nucleus (AGN) phases is not clear. We analyze a sample of low-mass early-type galaxies known to be in the process of migrating from the blue cloud to the red sequence via an AGN phase in the green valley. Using deeper imaging from SDSS Stripe 82, we show that the fraction of objects with major morphological disturbances is high during the early starburst phase, but declines rapidly to the background level seen in quiescent early-type galaxies by the time of substantial AGN radiation several hundred Myr after the starburst. This observation empirically links the AGN activity in low-redshift early-type galaxies to a significant merger event in the recent past. The large time delay between the merger-driven starburst and the peak of AGN activity allows for the merger features to decay to the background and hence may explain the weak link between merger features and AGN activity in the literature.
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