Tagged Questions

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Legendrian isotopy

I think the definition of Legendrian isotopy is that you can find an isotopy of the ambient manifold such that it takes a Legendrian knot to another through Legendrian knots. What I cannot figure out ...
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How to distinguish between knots and links based on knot diagrams/projections

I'm interested in the distinction between knots and links in $\mathbb{R}^3$/$S^3$. In particular, is there an algorithmic way (as in not by sight/intuition) that we can examine the arcs and crossings ...
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fibered knots in $S^3$

Given a fibered knot $k$ in $S^3$, we have the decomposition of $S^3$ as union of $M$ and $S^1\times D^2$, where M is a fiber bundle over $S^1$, with fiber $F$ such that its boundary is the knot $k$. ...
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Why is surgery along a framed link well defined?

Let $L=L_1 \cup L_2 \cup \cdots \cup L_n$ be a framed link in $S^3$. I want to perform the surgery along $L$ to get a new manifold $M$. By definition, to perform this surgery, I must perform the ...
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Coloring knot diagram obtained from colored one by applying crossing change to one crossing

Suppose $K_1$ is a knot diagram colored by a dihedral quandle $R_n$ of order $n$, By applying crossing change (exchanging over and under arcs) to one crossing in $K_1$, we obtain a new diagram let us ...
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double decker set in a surface-knot

Surface-knot is an embedded surface in $\Bbb{R}^4$. Project the surface in $\Bbb{R}^3$ gives the surface diagram with set of singularity points consists of double points, triple points and branch ...
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Relation between the braid group and the mapping class group of the plane

According to the following link, page 248, the braid group modulo its center is isomorphic to the mapping class group of the $N$-times punctured plane, i.e. $B_N/Z(B_N)\cong M_N(\mathcal(R)^2)$. Could ...
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Knot complement conjecture in solid tori

Has the knot complement conjecture been proven for knots in solid tori?
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How do you specify a link to a blind combinatorialist?

Regular projections of links look like graphs in the plane. So I'm wondering if it would be possible to specify a link up to isotopy with purely combinatorial data about this graph. If so, what kind ...
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Recommended books on knot invariants

I've been reading the books "An introduction to knot theory" by Lickorish and "Knots, Links, Braids and 3-Manifolds" by Prosolov and Sossinsky, and while both seem to me as good books, sometimes I'd ...
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Showing every knot has a regular projection using diff top

My question is: Can we use differential topology to prove that every smooth knot has a regular projection? Here is some background: Let $\gamma : S^1 \rightarrow \mathbb{R}^3$ be a smooth unit-speed ...
Uniqueness of Preferred Framing of a Solid Torus in $S^3$
One way to state my question tersely is: For a homeomorphism $f : S^1 \times \mathbb{D}^2 \rightarrow S^1 \times \mathbb{D}^2$, does $f|_{S^1 \times S^1}$ determine the isotopy class of $f$? This is ...