CONSTRUCTING DOUBLY-POINTED HEEGAARD DIAGRAMS COMPATIBLE WITH (1,1) KNOTS

A (1,1) knot K in a 3-manifold M is a knot that intersects each solid torus of a genus 1 Heegaard splitting of M in a single trivial arc. Choi and Ko developed a parametrization of this family of knots by a four-tuple of integers, which they call Schubert's normal form. This paper presents an algorithm for constructing a genus 1 doubly-pointed Heegaard diagram compatible with K, given a Schubert's normal form for K. The construction, coupled with results of Ozsváth and Szabó, provides a practical way to compute knot Floer homology groups for (1,1) knots. The construction uses train tracks, and its method is inspired by the work of Goda, Matsuda, and Morifuji.

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COMBINATORIAL KNOT FLOER HOMOLOGY AND DOUBLE BRANCHED COVERS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216513500144 ◽

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.

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Knots which admit a surgery with simple knot Floer homology groups

Algebraic & Geometric Topology ◽

10.2140/agt.2011.11.1243 ◽

2011 ◽

Vol 11 (3) ◽

pp. 1243-1256 ◽

Cited By ~ 1

Author(s):

Eaman Eftekhary

Keyword(s):

Floer Homology ◽

Homology Groups ◽

Knot Floer Homology

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An equivalent description of the lens space L(p,q) with p prime

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216520420055 ◽

2020 ◽

Vol 29 (10) ◽

pp. 2042005

Author(s):

Fengling Li ◽

Dongxu Wang ◽

Liang Liang ◽

Fengchun Lei

Keyword(s):

Complete System ◽

Lens Space ◽

Heegaard Splitting ◽

Prime Integer ◽

Heegaard Diagrams ◽

Heegaard Diagram ◽

Genus Formula ◽

Equivalent Description

In the paper, we give an equivalent description of the lens space [Formula: see text] with [Formula: see text] prime in terms of any corresponding Heegaard diagrams as follows: Let [Formula: see text] be a closed orientable 3-manifold with [Formula: see text] and [Formula: see text] a Heegaard splitting of genus [Formula: see text] for [Formula: see text] with an associated Heegaard diagram [Formula: see text]. Assume [Formula: see text] is a prime integer. Then [Formula: see text] is homeomorphic to the lens space [Formula: see text] if and only if there exists an embedding [Formula: see text] such that [Formula: see text] bounds a complete system of surfaces for [Formula: see text].

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SYMMETRIES AND ADJUNCTION INEQUALITIES FOR KNOT FLOER HOMOLOGY

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216512501040 ◽

2012 ◽

Vol 21 (10) ◽

pp. 1250104 ◽

Cited By ~ 1

Author(s):

BIJAN SAHAMIE

Keyword(s):

Floer Homology ◽

Homology Groups ◽

Knot Floer Homology

We derive symmetries and adjunction inequalities of the knot Floer homology groups which appear to be especially interesting for homologically essential knots. Furthermore, we obtain an adjunction inequality for cobordism maps in knot Floer homologies. We demonstrate the adjunction inequalities and symmetries in explicit calculations which recover some of the main results from [E. Eftekhary, Longitude Floer homology and the Whitehead double, Algebr. Geom. Topol. 5 (2005) 1389–1418] on longitude Floer homology and also give rise to vanishing results on knot Floer homologies.

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The knot Floer cube of resolutions and the composition product

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216520500066 ◽

2020 ◽

Vol 29 (03) ◽

pp. 2050006

Author(s):

Nathan Dowlin

Keyword(s):

Euler Characteristic ◽

Floer Homology ◽

Product Formula ◽

Knot Floer Homology ◽

Direct Sum ◽

Heegaard Diagram ◽

Composition Product ◽

The Relationship

We examine the relationship between the oriented cube of resolutions for knot Floer homology and HOMFLY-PT homology. By using a filtration induced by additional basepoints on the Heegaard diagram for a knot [Formula: see text], we see that the filtered complex decomposes as a direct sum of HOMFLY-PT complexes of various subdiagrams. Applying Jaeger’s composition product formula for knot polynomials, we deduce that the graded Euler characteristic of this direct sum is the HOMFLY-PT polynomial of [Formula: see text].

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