258 research outputs found
Mirror-Curves and Knot Mosaics
Inspired by the paper on quantum knots and knot mosaics [23] and grid
diagrams (or arc presentations), used extensively in the computations of
Heegaard-Floer knot homology [2,3,7,24], we construct the more concise
representation of knot mosaics and grid diagrams via mirror-curves. Tame knot
theory is equivalent to knot mosaics [23], mirror-curves, and grid diagrams
[3,7,22,24]. Hence, we introduce codes for mirror-curves treated as knot or
link diagrams placed in rectangular square grids, suitable for software
implementation. We provide tables of minimal mirror-curve codes for knots and
links obtained from rectangular grids of size 3x3 and px2 (p<5), and describe
an efficient algorithm for computing the Kauffman bracket and L-polynomials
[18,19,20] directly from mirror-curve representations
On sutured Floer homology and the equivalence of Seifert surfaces
We study the sutured Floer homology invariants of the sutured manifold
obtained by cutting a knot complement along a Seifert surface, R. We show that
these invariants are finer than the "top term" of the knot Floer homology,
which they contain. In particular, we use sutured Floer homology to distinguish
two non-isotopic minimal genus Seifert surfaces for the knot 8_3. A key
ingredient for this technique is finding appropriate Heegaard diagrams for the
sutured manifold associated to the complement of a Seifert surface.Comment: 32 pages, 17 figure
Grid Diagrams for Lens Spaces and Combinatorial Knot Floer Homology
Similar to knots in S^3, any knot in a lens space has a grid diagram from
which one can combinatorially compute all of its knot Floer homology
invariants. We give an explicit description of the generators, differentials,
and rational Maslov and Alexander gradings in terms of combinatorial data on
the grid diagram. Motivated by existing results for the Floer homology of knots
in S^3 and the similarity of the combinatorics presented here, we conjecture
that a certain family of knots is characterized by their Floer homology.
Coupled with work of the third author, an affirmative answer to this would
prove the Berge conjecture, which catalogs the knots in S^3 admitting lens
space surgeries.Comment: 27 pages, 8 figures; Expositional improvements, corrected
normalization of A grading in proof of Lemma 4.1
Trisections of 4-manifolds via Lefschetz fibrations
We develop a technique for gluing relative trisection diagrams of
-manifolds with nonempty connected boundary to obtain trisection diagrams
for closed -manifolds. As an application, we describe a trisection of any
closed -manifold which admits a Lefschetz fibration over equipped with
a section of square , by an explicit diagram determined by the vanishing
cycles of the Lefschetz fibration. In particular, we obtain a trisection
diagram for some simply connected minimal complex surface of general type. As a
consequence, we obtain explicit trisection diagrams for a pair of closed
-manifolds which are homeomorphic but not diffeomorphic. Moreover, we
describe a trisection for any oriented -bundle over any closed surface and
in particular we draw the corresponding diagrams for and using our gluing technique. Furthermore, we provide an
alternate proof of a recent result of Gay and Kirby which says that every
closed -manifold admits a trisection. The key feature of our proof is that
Cerf theory takes a back seat to contact geometry.Comment: 34 pages, 21 figure
Bordered Heegaard Floer homology and graph manifolds
We perform two explicit computations of bordered Heegaard Floer invariants.
The first is the type D trimodule associated to the trivial S^1 bundle over the
pair of pants P. The second is a bimodule that is necessary for self-gluing,
when two torus boundary components of a bordered manifold are glued to each
other. Using the results of these two computations, we describe an algorithm
for computing HF-hat of any graph manifold.Comment: 59 pages, 21 figures, new version corrects typos and adds a short
discussion of grading
Computing HF^ by factoring mapping classes
Bordered Heegaard Floer homology is an invariant for three-manifolds with
boundary. In particular, this invariant associates to a handle decomposition of
a surface F a differential graded algebra, and to an arc slide between two
handle decompositions, a bimodule over the two algebras. In this paper, we
describe these bimodules for arc slides explicitly, and then use them to give a
combinatorial description of HF^ of a closed three-manifold, as well as the
bordered Floer homology of any 3-manifold with boundary.Comment: 106 pages, 46 figure
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