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Results tagged with knot-theory
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user 36108
Knot theory is dealing with embedding of curves in manifolds of dimension 3. A knot is a single circle embedded in the affine space of dimension 3 as a smooth curve not crossing itself. Many knot invariants are known and can be used to distinguish knots.
7
votes
Accepted
Generating ribbon diagrams for knots known to be ribbon knots
I think Kawauchi's book has tables that include ribbon diagrams, but I don't have a copy with me. Look at
Livingston and Cha . It is not hard to get a ribbon disk from this diagram: add a handle bet …
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6
votes
Is there any analogs of Vassiliev invariants in higher dimensions?
Sorry for taking so long to respond. Theo's answer is fairly complete. I mentioned the question to Masahico who reminded me that there are results of Kanenobu and others about finite type invariants f …
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3
votes
Accepted
Is there a combinatorial version of PL ambient isotopy in dimension $>3$?
Unfortunately, this is not quite an answer. So for knotted surfaces in 4-space, there is Roseman's Theorem. I think that the context of Dennis's proof is in the smooth category. Or certainly the proof …
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3
votes
Accepted
Application of a quandle cocycle invariant for virtual knots
To find the answer to your question, I would look through papers of Sam Nelson and his students, all of which are available on the arXiv. It is true that most of his work has to do with using quandles …
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7
votes
Is every virtual knot group an HNN extension?
I don't have a full answer to your question, but a hint. Any virtual knot group is the fundamental group of a knotted torus in 4-space. Constructing the torus from the virtual diagram is easy. The con …
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4
votes
Higher order quandle
For any codimension 2 embedding that is locally flat, there is a notion of the fundamental rack (not every element is idempotent). In Euclidean space, there is a fundamental quandle. It is defined as …
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7
votes
0
answers
362
views
Is the category of tangles that includes, X, Y, and Lambda a free Frobenius braided category?
Consider the category whose objects are non-negative integers that are represented as dots along a line, and whose morphisms are generated by $X$---positive crossing, $\bar{X}$ --- negative crossing, …
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8
votes
Higher order quandle
The definition of strict 2-quandle and examples thereof has not be written down in a public forum yet. Crans, Elhamdadi, Saito, and I have a notion and examples. I think that Crans spoke about the ide …
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7
votes
0
answers
319
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What is the historical connection between Zeeman's twist spinning and Fox's Examples?
Both Ralph Fox and (at that time, yet to be knighted) Sir Christopher Zeeman attended the 1961 Georgia topology conference. Fox's paper from that conference was his seminal work, "A Quick Trip through …
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6
votes
Visualize Fourth Homotopy Group of $S^2$
Let me expand upon jc's comment above. For convenience, I'll pass to the stable homotopy group for a little while. The first stable stem is $\pi_1^s = {\mathbb Z}/(2)$. A representative class is the …
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3
votes
How to motivate the skein relations?
In the case of the Jones Polynomial, the R matrix that comes from the bracket relation really is a matrix. Consequently, it satisfies a polynomial equation. That polynomial equation can be thought of …
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4
votes
Can surfaces be interestingly knotted in five-dimensional space?
I always thought that Haefliger's result applied in dimensions higher than 5, but in skimming the manuscript, it looks like 5 might be a critical case. Please read at page 404 and 405 carefully.
To …
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7
votes
Proof of the Reidemeister theorem
Reidemeister's proof involves a single move: replacing 2 (or 1) edges of a triangle with the other edge (edges). It is in the English translation of his book.
I don't know if you can ferret out the …
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5
votes
Embedded ribbons and regular isotopy
A diagram is drawn in the plane. Restrict to knots (not links). Orient the curve, & associate to each crossing a (+/-) via a Right hand rule: palm along over-crossing with pinky pointing towards orien …
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8
votes
5
answers
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Braided Monoidal 2-categories with duals
Which categorifications give explicit braided monoidal 2-categories with duals?
This question is in response to Ben Webster's questions in recent history. The point is that given a braided monoidal 2 …
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