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Results tagged with gt.geometric-topology
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user 13119
Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).
7
votes
Accepted
Kirby diagram of the complement of a subhandlebody of a smooth closed 4-manifold
When the decomposition of $X$ has no 3-handles, this is often feasible. The trick is to turn the handle decomposition of $X$ upside down (see Example 5.5.5 in Gompf and Stipsicz's 4-manifolds and Kirb …
- 10.9k
28
votes
Accepted
The Jones polynomial at specific values of $t$
The evaluation of the Jones polynomial at $e^{i\pi/3}$ is related to the number of 3-colourings $tri(K)$ of $K$ (see also here) as well as to the topology of the branched double cover $\Sigma(K)$:
$$t …
- 10.9k
5
votes
Accepted
Why is this Brieskorn manifold a rational homology sphere?
The key is pointed out by HJRW in his comment: there's a missing piece in your explanation, which is the genus of the base $B$ of the Seifert fibration. Némethi writes:
$$
2g-2 = (n−2)A/a−\sum q_i,
$$ …
- 197
7
votes
Accepted
If a lattice can be embedded into $\mathbb Q^n,\langle-1\rangle^n$, then can it be embedded ...
As Ian Agol points out in his comment, $-E_8$ is an example of a lattice that embeds in $\mathbb{Q}^8$ but not in $\mathbb{Z}^n$ for any $n$.
The embedding in $\mathbb{Q}^8$ is given explicitly in Con …
- 12.9k
1
vote
Necessary condition for invertible knot concordance from both ends
No, this is not true: any doubly-slice knots gives a counterexample to your statement.
A knot $K$ is doubly-slice if there is an unknotted 2-sphere $F$ in $S^4$ such that $(S^3,K)$ is diffeomorphic to …
- 10.9k
7
votes
$3$-manifold that is a surgery on a knot
About your third question:
Are there any partial results for branched double covers of knots?
The Montesinos trick tells you that if $K$ has unknotting number 1, then its double cover is $\pm\frac{\ …
- 10.9k
5
votes
Euler number of a Seifert bundle as a generalization of an Euler number of a circle bundle o...
Why is the Euler number of a Seifert bundle a "natural" generalization of a circle bundle over a surface?
I can see at least three reasons:
A circle bundle is a Seifert manifold with no singular fi …
- 10.9k
10
votes
Presentations of exotic 4-manifolds
I guess that this is as explicit and low-tech as it gets: if $X$ is a K3 surface (i.e. a non-singular quartic hypersurface in $\mathbb{CP}^3$, with the complex orientation), then $X \# \overline{\math …
- 10.9k
7
votes
Generalization of the sphere theorem in dimension at least 4
Aru Ray and Danny Ruberman wrote a paper (here the arxiv version) about Dehn's lemma in dimension 4. From the abstract:
We investigate certain 4-dimensional analogues of the classical 3-dimensional D …
- 10.9k
7
votes
Detecting a "bad map" in Fintushel-Stern knot surgery
There is nothing inherently "bad" with other choices. My guess is that Fintushel and Stern chose this identification for three reasons: first, they can give a nice formula for how the Seiberg–Witten i …
- 10.9k
7
votes
Accepted
Picture of the isotopy class of a degree $d$ smooth complex curve
This might be a bit annoying to justify (see below), but I'm quite sure that the surface you're looking for is constructed as follows. (And in any case this is too long for a comment.)
Start with the …
- 10.9k
7
votes
Knot diagrams, sets of moves and equivalence relations
One classical example of such move is Conway mutation, which falls into the category of tangle replacement, as Qiaochu Yuan mentioned in his comment. There's a very famous pair of mutants, the Kinoshi …
- 4,713
9
votes
Accepted
Are two slice surfaces with minimal genus isotopic?
The question is very loaded and the question would almost require a survey...
Anyway, the answer to your questions is mostly no. Let $S \subset S^4$ be a 2-knot (i.e. an embedded 2-sphere), $p \in S$ …
- 10.9k
6
votes
Accepted
A Mazur manifold bounded by $\Sigma(2,3,13)$
I don't know where you took the picture from, but that link is not symmetric (and it likely does not describe the correct manifold).
Indeed, the 0-framed component of the link you drew is a trefoil: i …
- 10.9k
10
votes
Accepted
Two surfaces in a 4-manifold whose algebraic intersection number is zero
Yes, this can be done by tubing one surface along the other.
Suppose that you have two intersection points $p_+, p_- \in \Sigma_1 \cap \Sigma_2$ of opposite signs. Suppose also that $\Sigma_1$ and $\S …
- 10.9k