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5

The usual(?) Humphries generators for the mapping class group are Dehn twists about non-separating curves, and thus all conjugate. I believe something similar ought to be true about Out$(F_n)$, the outer automorphism group of a free group of rank $n$. The Nielsen generators are not all conjugate (some have finite order while others do not), but I would be ...

5

Given the mapping class group of a closed surface, there are many elements whose normal closure is the full group. See this article by Lanier and Margalit for more details.

3

Magma has a command $\mathtt{LowIndexNormalSubgroups}$ which can be used for this calculation. It took about 86 minutes to complete on this example, which I think is a little quicker than what you did. The code for this was written several years ago by a student of mine, and I know that a student of Alice Niemeyer has written an implementation of the same ...

3

This is rather a collection of hints, based on my earlier comments above: 1) Instead of using AllSmallGroups, you may find enumerating groups one by one more informative and convenient - see the Carpentries-style lesson on GAP, in particular this episode about the search in the Small Groups Library. 2) If you are only interested in the orders of all ...

1

One of the oldest examples of nontrivial links with zero linking number is the Whitehead Link. There are many other examples. If your wish is to classify, say, 2-component, links where each component is an unknot, by some simple numerical invariant (like the linking number), then there is no such classification. (One can even make this statement precise ...

1

Admittedly, the notation is confusing at first glance. However, I believe that earlier in the passage, this is sufficiently defined as Each crossing then corresponds to a linear expression given by: $a_i + a_j \Leftrightarrow 2ak$ , where $a_i$ and $a_j$ are the two understrands of the crossing and $a_k$ is the overcrossing. The $a_i$'s represent ...

1

I asked Adams at a talk earlier this year if it's still true that we don't know the (exact) hyperbolic volume of a single knot. His answer was *yes." --Ken Perko

1

The deeper meaning is that, at least for prime $n$, a Fox $n$-coloring is a homomorphism $\pi_1(S^3 \setminus K) \rightarrow D_n$ mapping meridians to reflections, where $D_n$ is the dihedral group. To see this, take a rotation $r \in D_n$ of order $n$ and any reflection $s$. Then you interpret the color $k \in \mathbb Z/n$ by the reflection $r^ks \in D_n$. ...

1

I first want to express my hesitation at answering the question the way you pose it here. Intuition is extremely valuable, but it can lead you astray if you are not careful. Nonetheless, I understand that you want an intuitive proof that shows the trefoil is not the unknot. The problem comes from the fact that there is no way, in general, to be sure we ...

1

Either I need some clarification or I have a counterexample. In the picture below I show an example of a disk $D$ with boundary the unknot that has an embedded arc $\ell$ with a fully twisted neighborhood ($D$ itself). Am I misunderstanding "fully twisted"? If we ask that $\ell$ is instead a simple closed curve, I think that there will be self-intersections....

1

For every $n\geq2$, Denis Osin constructed an uncountable class of finitely generated groups each of which contains precisely $n$ conjugacy classes of elements. Therefore, the $n-1$ non-trivial conjugacy classes generate the group, as they are the entire group, minus the identity! This is Corollary 1.4 of the paper Osin, Denis. "Small cancellations over ...

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