scholarly journals QUASI-ALTERNATING MONTESINOS LINKS

2009 ◽  
Vol 18 (10) ◽  
pp. 1459-1469 ◽  
Author(s):  
TAMARA WIDMER
Keyword(s):  
Open Problem ◽  
Floer Homology ◽  
Natural Generalization ◽  
Khovanov Homology ◽  
Knot Floer Homology ◽  
Interesting Open Problem ◽  
Alternating Links

The aim of this article is to detect new classes of quasi-alternating links. Quasi-alternating links are a natural generalization of alternating links. Their knot Floer and Khovanov homology are particularly easy to compute. Since knot Floer homology detects the genus of a knot as well as whether a knot is fibered, characterization of quasi-alternating links becomes an interesting open problem. We show that there exist classes of non-alternating Montesinos links, which are quasi-alternating.

scholarly journals A note on quasi-alternating Montesinos links

2015 ◽  
Vol 24 (09) ◽  
pp. 1550048 ◽  
Author(s):  
Abhijit Champanerkar ◽  
Philip Ording
Keyword(s):  
Recent Work ◽  
Floer Homology ◽  
Khovanov Homology ◽  
Joint Work ◽  
Knot Floer Homology ◽  
Rational Tangles ◽  
Heegaard Floer ◽  
Alternating Links

Quasi-alternating links are a generalization of alternating links. They are homologically thin for both Khovanov homology and knot Floer homology. Recent work of Greene and joint work of the first author with Kofman resulted in the classification of quasi-alternating pretzel links in terms of their integer tassel parameters. Replacing tassels by rational tangles generalizes pretzel links to Montesinos links. In this paper we establish conditions on the rational parameters of a Montesinos link to be quasi-alternating. Using recent results on left-orderable groups and Heegaard Floer L-spaces, we also establish conditions on the rational parameters of a Montesinos link to be non-quasi-alternating. We discuss examples which are not covered by the above results.

Properties of Littlewood-Paley sets

1989 ◽  
Vol 105 (3) ◽  
pp. 485-494 ◽  
Author(s):  
Kathryn E. Hare ◽  
Ivo Klemes
Keyword(s):  
Open Problem ◽  
Necessary Conditions ◽  
Combinatorial Characterization ◽  
Interesting Open Problem ◽  
Thin Sets

AbstractWe give a combinatorial characterization of the known examples of intervals of ℤ having the Littlewood-Paley (LP) property. This leaves an interesting open problem. We also note that certain previously known necessary conditions are equivalent, and give two examples of intervals which are not LP but whose endpoints form thin sets (e.g. Sidon or Λ(p) for all p < ∞).

scholarly journals Khovanov homology and knot Floer homology for pointed links

2017 ◽  
Vol 26 (02) ◽  
pp. 1740004 ◽  
Author(s):  
John A. Baldwin ◽  
Adam Simon Levine ◽  
Sucharit Sarkar
Keyword(s):  
Spectral Sequence ◽  
Floer Homology ◽  
Khovanov Homology ◽  
Knot Floer Homology ◽  
Text Page ◽  
Text Component

A well-known conjecture states that for any [Formula: see text]-component link [Formula: see text] in [Formula: see text], the rank of the knot Floer homology of [Formula: see text] (over any field) is less than or equal to [Formula: see text] times the rank of the reduced Khovanov homology of [Formula: see text]. In this paper, we describe a framework that might be used to prove this conjecture. We construct a modified version of Khovanov homology for links with multiple basepoints and show that it mimics the behavior of knot Floer homology. We also introduce a new spectral sequence converging to knot Floer homology whose [Formula: see text] page is conjecturally isomorphic to our new version of Khovanov homology; this would prove that the conjecture stated above holds over the field [Formula: see text].

Classical homological invariants are not determined by knot Floer homology and Khovanov homology

2016 ◽  
Vol 25 (07) ◽  
pp. 1650039
Author(s):  
Jae Choon Cha ◽  
Toshifumi Tanaka
Keyword(s):  
Floer Homology ◽  
Khovanov Homology ◽  
Knot Floer Homology ◽  
Homological Invariants

We illustrate that there are knots for which Heegaard knot Floer homology and Khovanov homology are identical but the Alexander module and torsion invariants differ. The examples are certain symmetric unions. We also give examples of similar flavor, concerning the Kauffman and Q-polynomials in place of the classical homological invariants. This shows there are nonmutant knots with the same knot Floer and Khovanov homology.

scholarly journals COMPUTATIONS OF HEEGAARD-FLOER KNOT HOMOLOGY

2012 ◽  
Vol 21 (08) ◽  
pp. 1250075 ◽  
Author(s):  
JOHN A. BALDWIN ◽  
WILLIAM D. GILLAM
Keyword(s):  
Floer Homology ◽  
Khovanov Homology ◽  
Combinatorial Approach ◽  
Close Attention ◽  
Knot Floer Homology ◽  
Knot Homology ◽  
Heegaard Floer

We compute the knot Floer homology of knots with at most 12 crossings, as well as the τ invariant for knots with at most 11 crossings, using the combinatorial approach described by Manolescu, Ozsváth and Sarkar. We review their construction, giving two examples that can be workout out by hand, and we explain some ideas we used to simplify the computation. We conclude with a discussion of knot Floer homology for small knots, and we formulate a conjecture about the behavior of knot Floer homology under mutation, paying especially close attention to the Kinoshita–Terasaka knot and its Conway mutant. Finally, we discuss a conjecture of Rasmussen on relationship between Khovanov homology and knot Floer homology, and observe that it is consistent with our calculations.

scholarly journals Compact Differences of Composition Operators on Bergman Spaces Induced by Doubling Weights

2021 ◽  
Author(s):  
Bin Liu ◽  
Jouni Rättyä ◽  
Fanglei Wu
Keyword(s):  
Bergman Space ◽  
Open Problem ◽  
Lebesgue Space ◽  
Composition Operators ◽  
Bergman Spaces ◽  
Weighted Bergman Space ◽  
Carleson Measures ◽  
Doubling Weights ◽  
The Way

AbstractBounded and compact differences of two composition operators acting from the weighted Bergman space $$A^p_\omega $$ A ω p to the Lebesgue space $$L^q_\nu $$ L ν q , where $$0<q<p<\infty $$ 0 < q < p < ∞ and $$\omega $$ ω belongs to the class "Equation missing" of radial weights satisfying two-sided doubling conditions, are characterized. On the way to the proofs a new description of q-Carleson measures for $$A^p_\omega $$ A ω p , with $$p>q$$ p > q and "Equation missing", involving pseudohyperbolic discs is established. This last-mentioned result generalizes the well-known characterization of q-Carleson measures for the classical weighted Bergman space $$A^p_\alpha $$ A α p with $$-1<\alpha <\infty $$ - 1 < α < ∞ to the setting of doubling weights. The case "Equation missing" is also briefly discussed and an open problem concerning this case is posed.

scholarly journals Quantum Mean-Field Games with the Observations of Counting Type

Games ◽  
2021 ◽  
Vol 12 (1) ◽  
pp. 7
Author(s):  
Vassili N. Kolokoltsov
Keyword(s):  
Game Theory ◽  
Nash Equilibrium ◽  
Open Problem ◽  
Existence And Uniqueness ◽  
Existence Of Solutions ◽  
Mean Field ◽  
Mean Field Games ◽  
Quantum Game ◽  
Quantum Games ◽  
Interesting Open Problem

Quantum games and mean-field games (MFG) represent two important new branches of game theory. In a recent paper the author developed quantum MFGs merging these two branches. These quantum MFGs were based on the theory of continuous quantum observations and filtering of diffusive type. In the present paper we develop the analogous quantum MFG theory based on continuous quantum observations and filtering of counting type. However, proving existence and uniqueness of the solutions for resulting limiting forward-backward system based on jump-type processes on manifolds seems to be more complicated than for diffusions. In this paper we only prove that if a solution exists, then it gives an ϵ-Nash equilibrium for the corresponding N-player quantum game. The existence of solutions is suggested as an interesting open problem.

scholarly journals COMBINATORIAL KNOT FLOER HOMOLOGY AND DOUBLE BRANCHED COVERS

2013 ◽  
Vol 22 (06) ◽  
pp. 1350014
Author(s):  
FATEMEH DOUROUDIAN
Keyword(s):  
Floer Homology ◽  
Combinatorial Proof ◽  
Branched Covers ◽  
Knot Floer Homology ◽  
Branched Cover ◽  
Heegaard Diagram

Using a Heegaard diagram for the pullback of a knot K ⊂ S3 in its double branched cover Σ2(K), we give a combinatorial proof for the invariance of the associated knot Floer homology over ℤ.

Casson-Lin's Invariant and Floer Homology

1997 ◽  
Vol 06 (06) ◽  
pp. 851-877 ◽  
Author(s):  
Weiping Li
Keyword(s):  
Euler Number ◽  
Floer Homology ◽  
Natural Generalization ◽  
Conjugacy Classes ◽  
Irreducible Representations ◽  
Representation Space ◽  
Integral Homology

Casson defined an invariant which can be thought of as the number of conjugacy classes of irreducible representations of π1(Y) into SU(2) counted with signs, where Y is an oriented integral homology 3-sphere. Lin defined a similar invariant (the signature of a knot) for a braid representative of a knot in S3. In this paper, we give a natural generalization of Casson-Lin's invariant. Our invariant is the symplectic Floer homology for the representation space of π1(S3 \ K) into SU(2) with trace-zero along all meridians. The symplectic Floer homology of braids is a new invariant of knots and its Euler number is the negative of Casson-Lin's invariant.

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