Questions tagged [knot-invariants]

For properties of knots that remain unaffected by Reidmaster moves

180 questions
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How to show that $\langle a,b \mid aba^{-1}ba = bab^{-1}ab\rangle$ is not Abelian?

I'd like to show that $$G = \langle a,b \mid aba^{-1}ba = bab^{-1}ab\rangle$$ is non-Abelian. I have tried finding a surjective homomorphism from $G$ to a non-Abelian group, but I haven't found one....
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Alexander polynomial of a knot vs Alexander polynomial of a knot exterior.

If I use SnapPy to compute the Alexander polynomial of the link in the picture, I get $$t_1^2 - t_1 + 1$$ which is just the Alexander polynomail of the trefoil. But when I compute the Alexander ...
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Virtual knot diagrams equivalent by second Reidemeister moves

Is there a way to decide if two virtual knot diagrams are equivalent using only second Reidemeister moves and virtual moves (i.e. the Reidemeister moves where at least one crossing is virtual)? Some ...
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Homology for virtual knot diagram

I am studying now the article of V. O. Manturov about free knots ("Free Knots and Parity", https://arxiv.org/pdf/0912.5348.pdf). I got stuck on the section "Parity as homology" (p.6), at the homology ...
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Virtual diagrams from oriented Gauss codes

Suppose that $D_1$ and $D_2$ are virtual diagrams of oriented virtual knots with the same oriented Gauss code (each crossing in the code contains the following information: crossing number, over/under,...
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Reshetikhin-Turaev Invariant of Manifolds

The Reshetikhin-Turavev construction comes with an invariant that is sometimes called the Reshetikhin-Turaev Invariant. I'm currently attempting to wrap my head around this construction but was ...
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I am reading this paper. Near the middle of page 2 the author seems to state that a framed link is just a ribbon graph. Is that an accurate statement?
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Stick number of Trefoil

So it is well know that the stick number (i.e. the minimum number that is need to make a knot out of -not necessarily of the same length- sticks) of every non-trivial knot is above six, with only the ...
Let us consider the two trefoils with Gauss codes $(1U+,2O+,3U+,1O+,2U+,3O+)$ and $(1O-,2U-,3O-,1U-,2O-,3U-)$ respectively ( $O/U$ - over/under, $-/+$ - negative/positive crossing type). These knots ...