# Tag Info

Accepted

### Does a compact contractible metric space have a point that is fixed by all isometries?

There are finite groups that act smoothly on a disk without a global fixed point. You can arrange the metric to be isometric, e.g. via the Mostow-Palais embedding theorem, which equivariantly and ...
• 26.7k
Accepted

### Coefficient of the top Pontryagin class in $L$-genus

The coefficient of $p_k$ is given by $$2^{2n}(2^{2n-1}-1)\frac{B_n}{(2n)!} = \zeta(2n)\frac{2^{2n}-2}{\pi^{2n}},$$ see e.g. Appendix A of this older version of Weiss (warning: for Weiss, the ...
• 7,432
Accepted

### How wild can an open topological 3-manifold be if it has a compact quotient?

The possible universal covers of closed 3-manifolds are $S^3-C$, where $|C|=0, 1, 2$ or $C$ is a tame Cantor set, corresponding to the space of ends of the fundamental group as you suspect. This ...
• 61.2k
Accepted

### Is there a geometric interpretation of the second derivative of the Alexander polynomial at $1$?

Given a knot in $S^3$, think of it as an embedding $$f : S^1 \to S^3.$$ The configuration space of $5$ distinct points in $S^3$ is denoted $C_5(S^3)$, this is a $15$-dimensional manifold and it ...
• 40.6k
Accepted

### Symmetries of contractable subsets of $\Bbb R^n$

I posted a refined/generalized question Does a compact contractible metric space have a point that is fixed by all isometries? and received an answer that contained all essential ingredients to ...
• 9,471
Accepted

• 22.3k
Accepted

### Does every simply connected, orientable, non-compact, 3-manifold embed in $\mathbb{R}^3$?

When the manifold is the universal cover of a compact $3$-manifold $M$ (to begin with, lets say without boundary) then you construct the embedding by hands, using geometrization. In your question let'...
• 40.6k
Accepted

### A ribbon presentation for a torus knot

As this composite knot is an example of a symmetric union of knots I suggest that you read my paper "The search for nonsymmetric ribbon knots" (see for instance Figures 1 and 4 and the ...

### Reference for an easy lemma on homeomorphisms of connected manifolds

In [Ancel and Bellamy, On homogeneous locally conical spaces, Fund. Math. 241 (2018), no. 1, 1–15] it is shown that every homogeneous locally conical connected separable metric space that is not a $1$-...
• 26.7k
Accepted

### Is there any genus one knot other than $5_2$ with Alexander polynomial $2t^2-3t+2$?

Ian Agol, in the comments says: Yes, there should be plenty. Think of the Seifert surface for the 5_2 knot as a disk with two strips (1-handles) attached. By tying knots into the strips (with zero ...

### Alexander invariant of torus knot

Putting everything together, and with the big help of "Knots" by Burde, I found the following solution. Let $f_n(t)\colon= 1+x+\cdots+x^{n-1}$, we note that $\frac{t^m-1}{t-1}=f_m(t)$. @...

### Is there any genus one knot other than $5_2$ with Alexander polynomial $2t^2-3t+2$?

Maybe a different approach but having the same flavor: Take a (untwisted) Whitehead double of any nontrivial knot $K$. Denote such a double knot $WD(K)$. Construct a satellite knot using $5_2$ knot as ...
• 783
Accepted

### Is every finitely generated classical Schottky group quasifuchsian?

Yes - every Schottky group is quasi-fuchsian. See Lemma 1 of Chuckrow's paper "On Schottky Groups with Applications to Kleinian Groups" published in Annals of Mathematics, 1968. The ...
• 18.5k

### Local complexity of triangulations

Theorem 3.37 of Florian Frick's thesis gives a sketch of the proof of statement in the question for smooth manifolds. At first sight, it only uses the Whitney embedding theorem and transversality, ...
• 7,432
Accepted

• 18.5k
1 vote

### fills a given polygon with a few types of given primitives

As Sam Nead suggested: Demaine, Erik D., and Martin L. Demaine. "Jigsaw puzzles, edge matching, and polyomino packing: Connections and complexity." Graphs and Combinatorics 23, no. 1 (2007):...
• 144k
1 vote

### Are two nonsimple closed geodesics in minimal position?

Assuming you are asking about geodesics in a hyperbolic surface, the answer is “yes”. This follows from the “bigon criterion”.
• 18.5k
1 vote

### Positive vs negative Dehn twist monoids

A general discussion may help clarify these points. There's a lot more to say (e.g. how the positive monoid gives Weinstein fillings, or how negative Dehn twists explicitly give overtwisted disks), ...
• 996
1 vote

### Fundamental groups of primitive non-algebraic compact Kähler manifolds

I am not sure how one checks primitivity for a compact Kähler manifold, this seems to be too delicate property. Intuitively speaking, very non-algebraic Kähler manifolds have very simple fundamental ...
• 758

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