Mathematics Stack Exchange
Mathematics Stack Exchange

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.

346 questions
39
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1 answer

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
  • 1,059
18
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2 answers

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?
user265163
12
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1 answer

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
  • 25,412
12
votes
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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9
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0 answers

Is this object a simpler Brunnian "rubberband" loop than those studied?

The standard configuration of Brunnian "rubberband" loops shows a series of unknots each bent into a U-shape, with their ends looped around the middle of the next unknot, as shown here (drawn by David Epstein) Brunnian rubberband loop object: This…
egmoen
  • 91
9
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2 answers

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
  • 193
9
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1 answer

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
  • 25,412
8
votes
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…
Jules Randolph
  • 91
8
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1 answer

Definition by degree and intersection number are equivalent (linking number). [repost]

I will here restate a question I asked earlier. It did not have much succes (probably by an incomplete introduction of the problem on my part). I am reading a paper by Ricca ( http://www.maths.ed.ac.uk/~aar/papers/ricca.pdf p.1338 on the bottom) on…
Tom Ultramelonman
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7
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1 answer

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.
  • 673
7
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1 answer

Why a torus knot is a prime knot?

Why a torus knot is a prime knot?
IBazhov
  • 545
7
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1 answer

Struggling to Show Alexander Polynomial is a Knot Invariant Using Skein Relations

For (one of) the books I am using to learn knot theory, the Alexander polynomial is defined by the skein relation, or the unknot has polynomial 1 and the relation $\Delta(L_+)-\Delta(L_-)+(t^{1/2}-t^{-1/2})\Delta(L_0) = 0$. Following this, the book…
junglekarmapizza
  • 171
7
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0 answers

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.…
Akiva Weinberger
  • 25,412
6
votes
1 answer

Are there examples of different knots with identical Jones polynomials and different Seifert Genus?

I'm wondering if its ever possible to find two non-isotopic knots which have identical jones polynomials but different seifert genus? Attempting to google for this I found this example of non-isotopic knots with identical jones polynomials. But in…
Sidharth Ghoshal
  • 18,359
6
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0 answers

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…
urbanog
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