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8 votes
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Is the number of prime factors of 3-manifolds obtained by Dehn surgery along a link with $N$ components in $S^3$ bounded from above?

For a given $N$, is the number of prime factors of 3-manifolds obtained by Dehn surgery along a link with $N$ components in $S^3$ bounded from above? The Two Summands Conjecture states that surgery ...
Arshak Aivazian's user avatar
7 votes
0 answers
130 views

Generalized Brunnian links

A Brunnian link of order $n$ is nontrivial link of $n$ rings that becomes a trivial link of $n-1$ rings if any ring is removed. They were classified up to link-homotopy by Milnor in 1954. This ...
Richard Stanley's user avatar
6 votes
0 answers
211 views

$\mathbb{Z}/2\mathbb{Z}$ coefficients in gysin sequence

I am reading the article "Signature of links" by Kauffman and Taylor. Here they show that it is possible to calculate the nullity of a link $L\subset S^3$ by knowing the first betti number of the ...
Diego95's user avatar
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5 votes
0 answers
328 views

Are there examples of different knots with identical Jones polynomials and different Seifert Genus?

I had asked this question on math.stackexchange 2 days back but came up empty handed so I wanted to ask it here. Are there known examples of $2$ non equivalent knots that have identical jones ...
Sidharth Ghoshal's user avatar
4 votes
0 answers
112 views

Coloured Jones polynomial of the mirror image of a multicomponent link

This question has been reposted from MathStackExchange It is well understood that the usual Jones polynomial of a knot or link can be related to the Jones polynomial of the mirror image of the knot/...
hopftype's user avatar
4 votes
0 answers
181 views

Borromean Lines of three $\mathbb{R}^1$ in $\mathbb{R}^3$ and analogous Milnor link invariants

It is know that Borromean rings can be detected by Milnor invariant $$ \bar{\mu}(\gamma_1,\gamma_2,\gamma_3)= \# (\Sigma_1 \cap \Sigma_2 \cap \Sigma_3)-\frac{1}{2}\sum_{I,J,K}\epsilon_{IJK} \sum_{\...
annie marie cœur's user avatar
3 votes
0 answers
57 views

What's the Milnor's link group for the trivial knot in a lens space?

For a link $L$ in a 3-manifold $Y$, Milnor's paper "Link Groups" https://link.springer.com/content/pdf/10.1007/BF01393902.pdf defined the link group as some quotient of $\pi_1(Y-L)$. If $L$ ...
Faniel's user avatar
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2 votes
0 answers
44 views

Link invariants on a thickened surface

Let $\Sigma$ be an oriented surface. I want to know about link invariants in $\Sigma\times [0,1]$. I already know the Ozawa polynomial introduced in this paper, but I couldn’t find any other than that....
AW.'s user avatar
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2 votes
0 answers
80 views

Composition of 3-braids to obtain braids with trivial closure

Given a 3-braid $b=\sigma_1\sigma_2^{-1}\sigma_1\sigma_2^{-1}\sigma_1$ (which has non-trivial closure), can we find a 3-braid $c$, which has trivial closure (closure results in any trivial knot or ...
Muqing Cao's user avatar
2 votes
0 answers
135 views

Möbius cross energy in $S^3$?

Let $\gamma_i$, $i=1,2$ be two loops in $\mathbb R^3$. The Möbius cross energy of the pair is defined by $$ E(\gamma_1, \gamma_2)=\iint_{S^1\times S^1}\frac{|\gamma'_1(u)|\cdot|\gamma'_2(v)|}{|\...
J. GE's user avatar
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2 votes
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Flat or linkless embeddings of graph with fixed projection

The problem of finding whether a given planar diagram of a graph, with over- and under-crossings, is a linkless embedding or not has unknown complexity (Kawarabayashi et al., 2010). My first question ...
Herng Yi's user avatar
  • 221
2 votes
0 answers
109 views

Compare two topologies: Three 2-tori inside $S^3 \times S^1 \# S^2 \times S^2$ glued from two different diffeomorphisms

We like to ask for the comparison of two topologies of three 2-tori inside the same 4-manifolds glued from two different diffeomorphisms (see the end). Given an embedded torus $T$ with trivial normal ...
wonderich's user avatar
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1 vote
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Mac Lane-like condition for intrinsically linked graphs?

If any embedding of your graph in 3-space has two cycles that are linked, then your graph is intrinsically linked (such as the Petersen graph). These graphs generalise non-planar graphs since for ...
ben macintosh's user avatar
0 votes
0 answers
181 views

How does the extra rope length of this link/tangle scale with the inner triangle size?

The symmetric chiral link made of three long intertwined/linked/tangled flexible ropes of radius 1 shown in the figure, whose 6 ends all lie in a plane at spatial infinity and which are pulled ...
Claudio's user avatar
0 votes
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Is the 3d writhe of ideal knots proportional to their smallest possible 2d writhe?

In a knot, the (two-dimensional) or 2d writhe is the sum of all positive crossings minus the sum of all negative crossings. The 2d writhe is always an integer. There is also, for each knot, a smallest ...
Claudio's user avatar
0 votes
0 answers
82 views

Is there any functoriality of Stallings' twists?

Suppose $L$ is an oriented link and $L'$ is a link obtained from the Stallings' twists. Are there any functoriality of the twist operation on the links, such as functoriality between (1) homology ...
jhbaik's user avatar
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