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Results tagged with gt.geometric-topology
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user 9417
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).
38
votes
3
answers
3k
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Why are there no wild arcs in the plane?
On math.stackexchange it was asked whether all arcs in the plane are ambient-isotopic. I suggested that one could prove this by appealing to the Schönflies theorem, which you can do as long as you can …
- 4,898
23
votes
4
answers
3k
views
Utility of virtual knot theory?
Virtual knot theory is an interesting generalization of knot theory in which ``virtual" crossings are allowed. See Kauffman's Virtual Knot Theory for an introduction. Greg Kuperberg gave a nice topolo …
- 4,898
19
votes
1
answer
2k
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Propagation of an error in the LMO invariant? (Revision: I don't think LMO is wrong!)
Edit: I think LMO is correct. Massuyeau has a nice explanation here.
Edit: Renaud Gauthier has retracted the claim of an error in the foundations of the LMO construction, and has withdrawn both prepri …
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15
votes
Intuition behind Alexander duality
I like to think of Alexander duality in terms of linking numbers of submanifolds (or, in general, k cycles). This is one way to define the pairing you are looking for. In general, consider a $k$-cycle …
- 4,898
13
votes
Whitehead doubles of any knots
Ian's answer is very elegant, but in case you're looking for a more computational approach, you could use the Seifert form. Namely, if you take a Seifert surface $\Sigma$ for a knot, look at the form …
- 4,898
13
votes
3
answers
739
views
Algorithm for detecting ribbon or slice links?
A link in $S^3$ is said to be slice if it bounds a collection of flat disks into the $4$-ball. Here "flat," means that there is a (locally) trivial normal bundle. This condition can be strengthened to …
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10
votes
Accepted
Knot theory without planar diagrams?
Here is an example close to my heart. We show that the second coefficient of the Conway polynomial can be defined by counting quadrisecants. (Collinearities of $4$ points on the knot.)
(source: rybu …
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9
votes
Accepted
Vassilliev invariants of knots and their cables
As I mentioned in a comment, for the degree $2$ invariant $v_2$ which is the coefficient of $z^2$ in the Conway Polynomial, we have that $v_2(K_{p,q})=av_2(K)+b$. If $K$ is the unknot, this implies th …
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8
votes
2
answers
1k
views
Quasi-Lie algebras in nature?
A Lie algebra over $\mathbb Z$ is defined to be an abelian group with a bracketing operation $[\cdot,\cdot]$, satisfying the Jacobi identity and the relation $[x,x]=0$ for every $x$. On the other hand …
- 4,898
7
votes
Accepted
rational cohomology of symmetric groups
Take the $n$-skeleton. It has trivial rational homology except possibly in degree $n$. Now add enough $n+1$-cells from the $n+1$-skeleton to kill this top homology. You won't have created any $n+1$-di …
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7
votes
Accepted
Linking number a complete invariant of link homotopy
The linking number is the same as the homology class that one component represents in the complement of the other. You can reduce any $2$-component link to a normal form by first homotoping one compon …
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6
votes
3
answers
381
views
Surgery along an arc connecting the components of a $2$-component link gives the unknot
Math Overflow seems to have a dearth of low dimensional topology, but this seems like an interesting question. Let $L$ be a $2$-component link in $S^3$. Suppose that there is a framed arc joining the …
- 4,898
6
votes
0
answers
273
views
Torsion in chord diagrams for 2-string links?
The abelian group of $k$-chord diagrams on a skeleton of two directed line segments (modulo the STU relation), $\mathcal A_k(\uparrow\uparrow)$, is known to have $2$-torsion when $k=5$. In fact, I kno …
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6
votes
Accepted
Boundary links and ribbon links.
The answer to your first question is no. There are non-ribbon boundary links whose components are unknotted! Indeed the Bing double of a knot is a boundary link with unknotted components, but it has r …
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3
votes
Fibered example of topologically slice knots
A common source of topologically slice knots are those with Alexander polynomial $1$. However these are not fibered. This follows from a classical result, that $2\mathrm{genus}(K) = \mathrm{deg}(\Delt …
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