All Questions
10 questions
14
votes
4
answers
1k
views
Obtain any 3-manifold from repeating surgeries on knots in $S^3$
In Witten's “QFT and Jones Polynomials” paper, page 383, it states that: "It is a not too deep result that every 3-manifold can be obtained from or reduced to $S^3$ (or any other desired 3-manifold) ...
- 755
12
votes
4
answers
1k
views
Elementary proof that knot complements are path-connected
The complement of any (topological) knot is path-connected. More precisely, if $K$ is a subset of $\mathbb{R}^3$ (or $S^3$) homeomorphic to $S^1$, then $\mathbb{R}^3\setminus K$ (or $S^3\setminus K$) ...
- 35.9k
10
votes
1
answer
207
views
The knot $K\subset \Bbb S^3$ is smoothly slice, but the disc $D\subset \Bbb D^4$ is only locally flat. Can $D$ be smoothed?
Suppose I am given a smoothly slice knot $K\subset\Bbb S^3$. But I am only given a locally flat disc $D\subset \Bbb D^4$ with boundary $K$.
Question: Is there a smooth disc $D'\subset\Bbb D^4$ with ...
- 13.6k
3
votes
1
answer
161
views
How to properly define a slice knot (or a locally flat disk)?
A knot $K\subset\Bbb S^3=\partial \Bbb D^4$ is said to be (topolopgically) slice if there is a locally flat disk $D\subset\Bbb D^4$ with $\partial D=D\cap \Bbb S^3=K$. As far as I understand, locally ...
- 13.6k
3
votes
1
answer
292
views
Can a closed trefoil appear as a space-time "cut" of an open trefoil?
An open trefoil is a trefoil tied in an infinitely long line. An open trefoil that is at rest in (flat) space, in space-time is a knotted hypersurface.
Different observers in space-time have ...
- 75
3
votes
0
answers
106
views
A link of four 2-tori $T^2$ in $S^3 \times S^1 \# S^2 \times S^2 \# S^2 \times S^2$
Step 1: We glue two sets of complement space of $D^2\times T^2$ out of the 4-sphere $S^4$, through their $T^3$ boundary to obtain a new 4-manifold:
$$(S^4 \smallsetminus D^2\times T^2) \cup (S^4 \...
- 10.5k
2
votes
1
answer
166
views
What is the most symmetric configuration of four 2-surfaces linked in $S^4$?
What are some of the most symmetric configurations of four 2-surfaces linked in the 4-dimensional sphere $S^4$?
To make a lower-dimensional analogy, recall that in 3-dimensional sphere $S^3$, we can ...
- 10.5k
1
vote
1
answer
91
views
When is a 2-bridge knot hyperbolic?
It is known that 2-bridge knots in $S^3$ can be classified by the Schubert form. My question is: which 2-bridge knots are hyperbolic? (Do we have a complete classification for hyperbolicity in 2-...
- 605
0
votes
1
answer
277
views
Are knot invariants topological invariants? [closed]
I am a bit confused about terminology considering topology and knot theory.
A topological invariant is considered to be a topological property that does not change under a homeomorphism of the space.
...
- 1,465
0
votes
1
answer
210
views
Questions on the proof Lemma 4.5 GTM 175, Lickorish
I am reading GTM 175 An introduction to knot theory by Lickorish and have some questions on the proof of Lemma 4.5 given.
For (a), it says "Suppose that $C$ is amongst the $n$ components of $F\...
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