Tagged Questions

For questions on knot theory, the study of mathematical knots

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Linking Number $0$ When One Knot is a Boundary

I am given two smooth knots $K$ and $L$ embedded in $\mathbb{R}^3$. I am also given that $K$ is the boundary of an oriented, compact surface disjoint from $L$, call it $S$. I must prove that the ...
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Introduction to subfactor theory

I have almost no knowledge about subfactor theory but I would like to understand what it is. As a self-learner, I do not know where to start. Could you suggest introductory text/paper/book to study ...
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Potential proof for the Slice-Ribbon conjecture (may be wrong).

Let $f:(D^2,S^1)\to(D^4,S^3)$ be a smooth embedding (so called a slice disk), and we set $M:=f(D^2)$. Then, is the restriction map $C^{\infty}(D^4)\to C^{\infty}(M)$ open map with relative to $C^2$ ...
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Ambient isotopy taking a polygon knot to another polygon knot

Let me define that a polygon knot means a knot K of which all point belongs to some line segment which is a subset of K. Let me ask if knot theory has some proof for that: For all pair ( K1, K2 ) ...
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Knots from the boundary of Möbius strips

A Möbius strip with one half twist has the unknot as its boundary. One with two half twists has a link of two unknots. One with three half twists has the trefoil knot as its boundary. Years ago, I ...
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figure-8 knot complement

The figure-8 knot seen as a 2-bridge knot with two maxima and two minima of the height function, has a complement in $S^3$ with one 0-handle,two 1-handles, two 2-handles and a 3-handle which cancels ...
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Are the generators of the braid group conjugates?

In his classic paper on Hecke algebra representations of braid groups from 1987, Vaughan Jones makes the claim that "the various generators $\sigma_i$ are all conjugate." How does one see this? I ...
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Are 3#4 and 3*#4composite knots isotopic?

all I found (on wolfram) that there is one composite knot with seven crossings and that is the 3#4. But is this really equivalent to 3*#4 i.e. a composite knot of trefoil with opposite chirality and ...
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Homology of knot exterior in general manifold for not null-homologus knot

I was trying to figure out the homology of the knot complement when $K$ is not a rational null-homologous knot ($[K]\neq 0\in H_1(X_K,\mathbb Q)$). We then know by half-lives half dies that the ...