Questions tagged [knot-invariants]

For properties of knots that remain unaffected by Reidemeister moves

Proving that two knots are not the same by proving that there is no injective homotopy from a knot to another is normally quite hard. Fortunately there are some properties of knot projections that remain invariant by Reidemeister moves, thus two knots with different invariants cannot be equivalent.

This tag is for questions concerning said properties.

279 questions
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Can I solve an integral (or other tough problem) by playing with knots?

I've seen that in calculating things in knot theory that involves a lot of hard looking integrals and matrices, even though the knots themselves appear fairly simple. So is there some way in which this works backwards, where I might have a really…
Kainui
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Laymans explanation of the relation between QFT and knot theory

Could someone give an laymans explanation of the relation between QFT and knot theory? What are the central ideas in Wittens work on the Jones polynomial?
12
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What is the degree of an n-fold branched cover over a trefoil?

The order-2 cyclic branched cover over a trefoil has degree 6, meaning the preimage of any point off the trefoil has cardinality six. (You can find a wonderful video of this here, made by Moritz Sümmermann.) The order-3 cyclic branched cover over a…
Akiva Weinberger
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12
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2 answers

Jones Polynomial from Statistical Mechanics

I've been told that, given a knot projection, there is a way of associating a statistical system in such a way that the partition function of the system corresponds to the Jones polynomial of the original knot. I have a rough understanding of how…
Mark B
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8
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Can the Borromean rings be unlinked if we allow each component to pass through itself?

Every knot and link can be untied if you allow it to pass through itself. (This is why the unknotting number is always finite.) This changes if you only allow each component of the link to pass through itself, prohibiting them from passing through…
Akiva Weinberger
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7
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Can information about a knot be recovered from the Jones Polynomial?

Suppose we know the Jones polynomial of some knot, but maybe not specifically which knot. Can any information about the knot be recovered just by knowing its Jones polynomial? Say, for example, the knot's unknotting number, or the minimal number of…
Felix Y.
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7
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Why a torus knot is a prime knot?

Why a torus knot is a prime knot?
IBazhov
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7
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Given a knot, what's the minimal genus of a torus the knot is embeddable on?

An n-embeddability definition appears towards the end of the section 5.1 Torus knots of the Knot book by C. C. Adams: A knot $K$ is an $n$-embeddable knot if $K$ can be placed on a genus $n$ standardly embedded surface without crossings, but $K$…
charlie
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6
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Is this knotted graph knotted?

Above, I've drawn a knotted 3-valent graph. I suspect that it's not isotopic to the "unknotted" version below, but I'm not sure. Is it? I know about fundamental groups of knot complements, but it seems like that's probably more work than necessary.…
6
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Homology of a submanifold complement

Let $X$ be a closed, connected, orientable smooth n-manifold and let $Y\subset X$ be a smooth closed m- submanifold. Express homology of the complement $H_*(X\smallsetminus Y;\mathbb{Z})$ in terms of (co)homology of the pair $(X,Y)$. Compute…
5
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Seifert surfaces for knots $6_1, 6_2, 6_3$.

I have been trying to calculate the genera of these knots, but the first step in doing so is to convert them into orientable knots by constructing Seifert surfaces for those knots. I started to do this using the algorithm here…
5
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2 answers

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. The context is that I would like to show that…
5
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1 answer

Can two different prime knots have a Dowker-Thistlethwaite code in common?

I was thinking about knot invariants and whether we could define an equivalence class on the set of all Dowker-Thistlethwaite codes for a knot, and whether said equivalence classes, combined with some indicator of chirality, would be a complete knot…
5
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2 answers

the evaluation of the Jones polynomial of an alternating link at $ t= -1 $.

I've been looking at some graph polynomials and I found a very nice relation between the famous Tutte polynomial of graphs and the no less famous Jones polynomial of links. Using this relation I was able to show, that for an alternating link $ L $…
Simo Sabak
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5
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Tight approximation of a Torus Knot length

Is there a simple formula for a tight approximation of the torus knot length ? (specifically a formula that does not involve integrals or any iterative procedures). The torus knot parameters are $(p, q, R, r)$ where $(p,q)$ are co-primes and $R$ is…
DolphinDream
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