9
votes
Is Rasmussen's s-invariant of a knot an invariant of a 4-manifold?
Lisa Piccirillo just posted a few days ago this paper on the arXiv. Corollary 1.3 asserts exactly that $s$ is not a 0-trace invariant of the knot.
- 10.9k
8
votes
Accepted
Heegaard Floer homology of a genus two Heegaard splitting of $S^3$
There are several things going on here, explained rather elliptically in the paper. Let me expand.
First, there's the question of which holomorphic annuli double-cover the disk. More precisely, ...
- 10.1k
6
votes
Accepted
How to use a Heegaard diagram to retrieve the original 3-manifold that it represents?
Short answer: The key is in the phrase "produce the original three-manifold that it represents?". The answer is you do not recover the original manifold. Instead the Heegaard splitting ...
- 28.2k
5
votes
Accepted
On Ozsváth and Szabó's branched covering description of holomorphic disks in symmetric products
They really mean to evaluate $\hat u$ on the $g$ points (with multiplicity) in $p^{-1}(z)$, so $u(z)=[\hat u(z_1),\ldots,\hat u(z_g)]$ where $p^{-1}(z)=\lbrace z_1,\ldots,z_g\rbrace$ (with possible ...
- 17.5k
5
votes
Heegard diagrams for three-manifolds
Chapter four of "Knots, Links, Braids and 3-Manifolds" by Prasolov and Sossinsky gives a highly readable (and nicely illustrated) introduction to three-manifolds via Heegaard splittings. ...
- 28.2k
4
votes
Accepted
Extending curves on a surface to a basis for its first homology satisfying intersection criteria
If I understand your question correctly, what you’re looking for is Lemma A.3 in my paper here.
- 44.8k
4
votes
Upsilon of an alternating knot
The existence of generators $x_i$ and $y_i$ follows from the fact that alternating knots are thin (i.e. their knot Floer homology is supported on a diagonal $M-A=\text{constant}$) and (the filtered ...
- 10.9k
3
votes
Topological type of complement of Heegaard curves in Heegaard surface $(\Sigma - \alpha - \beta)$
My standing assumption is that the $\alpha$ and $\beta$ systems of curves have been isotopes into minimal position (and thus $\Sigma - (\alpha \cup \beta)$ has no bigons).
There are two cases.
In the ...
- 28.2k
3
votes
Algebraic variations of the full knot Floer complex
A version of knot Floer homology defined over the ring $\mathbb{F}[U,V]/(UV)$ is roughly equivalent to the information which is called, in Holomorphic discs and knot invariants, in the statement of ...
- 155
3
votes
Heegard diagrams for three-manifolds
You are probably familiar with definitions and theorems. But I prefer to write those for completeness. And also excuse for a paint-like drawing. I hope that they will be useful.
A handlebody of genus $...
- 1,292
3
votes
Computation of \tau invariant
See Livingston's "Computations of the Ozsvath-Szabo knot concordance invariant" (https://arxiv.org/abs/math/0311036), Corollary 3.
Or see my thesis for a picture of Livingston's cobordism (p. 19, ...
- 890
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