Questions tagged [knot-invariants]
For properties of knots that remain unaffected by Reidemeister moves
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Understanding Milnor $\bar{\mu}$-Invariants
I find Milnor's $\bar{\mu}$-invariants a bit confusing. My understanding of their calculation is as follows:
draw the link diagram, label the arcs
write down Wirtinger presentation of the link group $...
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Unknotting number
How is the unknotting number a knot invariant? I mean, if I have two links which are ambient isotopic do they have the same unknotting number?
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Where can I find a proof that the crossing number of a knot/link is a knot invariant?
Where can I find a proof that the crossing number of a knot/link is a knot invariant?
I know that this is in fact a true statement when you consider that the presentation of the knot is the one with ...
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Computing the Cohomology of Knot Space vs Cohomology of Non-compact Knots
In Vassiliev's paper of the cohomology of knot spaces, he considers the long-knots or non-compact knots i.e. embedding of $S^1$ into $S^3$ that go through a fixed point.
He claims that this is meant ...
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Power series expansions and limits of knot invariants
I move the question here
Background(ignore this paragraph if you know finite type invariants well): Recall that a finite type invariant of degree $n$ is an invariant $V$ such that $V^{(n+1)}=0$ where $...
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Characteristic class detecting "upward-facing" surfaces
Let $\Sigma \subseteq \mathbb{R}^3$ be a smoothly embedded compact oriented surface with boundary. Let $\vec{n}: \Sigma \rightarrow \mathbb{R}^3$ be the field of unit normal vectors associated to the ...
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Proving that an algebraic (aborescent) link that has exactly one negative sign in its Conway notation has an almost alternating projection
I'm working on Exercise 5.32 in The Knot Book by Colin Adams, which asks to prove that an algebraic link that has exactly one negative sign in its Conway notation has an almost alternating projection.
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Software packages to compute finite type invariants of Polygonal Knots
Assume I have a polygonal knot, $K$, represented as its set of vertices $\{\mathbb{v}_i| \mathbb{v}_i\in\mathbb{R}^3\}_{i=1,...,n+1}$, where $n$ is significant, let's say $100<n<500$.
Which ...
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When is Morton-Franks-Williams inequality for knots strict?
I am reading Kawamuro's paper on Morton-Franks-Williams inequality (https://arxiv.org/abs/math/0509169). It says that a knot $K$ with braid index $b$ and maximal/minimal degrees of the variable $v$ ...
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Omitting the last relation in the Wirtinger presentation of a link group
In my knot theory class homework I encountered the following question:
Prove that for every link, when calculating the Wirtinger presentation of the fundamental group of its complement, you can ...
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Connection between different ways to calculate the knot determinant
I understand that there are multiple ways to calculate the knot determinant, one is through the Alexander polynomial, the other is by creating another matrix which uses the linesections and crossings, ...
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Chirality and Colored Jones Polynomial
It is well understood that the usual Jones polynomial of a knot or link can be related to the Jones polynomial of the mirror image of the knot/link by changing the variable $V_L(t) \to V_L(t^{-1})$. ...
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Confusion on the Definition of Isolated Chord
An isolated chord diagram is usually defined to be a chord diagram with a chord that doesn't intersect any other chord. But in this notes, it is defined to be a diagram with a chord that relates two ...
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What is the Jones Polynomial for the Borromean Link?
I was looking up the Jones Polynomial for a project I’m working on and came up with this equation from the knot atlas:
$$ -q^3-q^{-3}+3q^2+3q^{-2}-2q-2q^{-1}+4 $$
However, I know that when entering VL(...
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2-bridge knot with straightened strand
Apparently, every 2-bridge knot can be drawn such that of the four strands in the braid word, one strand remains straightened and is not crossing any of the other strands. Is there a general algorithm ...
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