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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).
6
votes
3
answers
381
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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
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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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 …
- 4,898
3
votes
Accepted
Handle slides homeomorphism
I'm too lazy to draw this right now, but I think I can describe it anyway. Consider a twisted band attached right next to a dual pair of untwisted bands. I'll write this $A\bar{A}BCBC$. Here the overb …
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2
votes
Accepted
Where can one find reference proving that Braid group induces isomorphism between punctured ...
The isomorphism of fundamental groups comes from a diffeomorphism of spaces: $D\times[0,1]\setminus B$ is diffeomorphic to the product of an n-punctured disk with $[0,1]$. To see this, note that you c …
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0
votes
generators of Out(F_n) and homology
You can cook up lots of normal subgroups by looking at any characteristic subgroup of the free group. For example, if $F^{(k)}$ is the $k$th term of the lower central series, there is a surjection
$$O …
- 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 …
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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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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
2
votes
Utility of virtual knot theory?
This paper by Chrisman and Manturov is as close as possible to an answer to my original question. From their introduction:
By classical knot theory we mean the study of knots and links in the $3$- …
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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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6
votes
0
answers
273
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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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2
votes
Accepted
Quasi-Lie algebras in nature?
Tom Goodwillie:
The homotopy groups of a (say, simply connected) space $X$ form a graded Lie algebra under Whitehead product, in which the even-dimensional part (which is actually the $\pi_n$ for …
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
13
votes
3
answers
739
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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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