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Questions tagged [knot-invariants]

For properties of knots that remain unaffected by Reidmaster moves

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Finding a presentation matrix with low dimension.

Let $R=\mathbb Z[t^{\pm}]$ and $M$ a finitely generated $R$-module. With $A$ a presentation matrix, i.e we have the following exact sequence $$ 0\rightarrow R^m\overset{A}{\rightarrow} R^n\rightarrow ...
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1answer
63 views

Using grid diagram to compute the Alexander polynomial

I have been reading the book 'Grid Homology for Knots and Links' (see https://web.math.princeton.edu/~petero/GridHomologyBook.pdf) - in Section 3.3 it provided a way to compute the Alexander ...
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32 views

winding number and Jordan-Schoenflies Theorem

I have a statement which is related to the smooth Jordan-Schoenflies Theorem. I can verify it in some simple cases. The setup is: Let there be a $C^1$ immersion $\gamma : S^1 \to \mathbb{R}^2$. ...
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1answer
57 views

Map of $\mathbb{R}^3-Knot \to S^1$

Reading Bachman's "A Geometric Approach to Differential Forms", in section 7.8.1 about the Lining Number invariant, I have stumbled upon the following assertion. Let the knot $K$ be defined as a (...
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44 views

Show Jones polynomial is unique

I am trying to show that if we define a new invariant of knots $W(L)$ which follows the same rules as the Jones polynomial $V(L)$, so that it has value 1 on the unknot and satisfies the Skein relation ...
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114 views

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 $n$ ...
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1answer
44 views

torus crossing directions & Conway polynomial

I know that the Conway polynomial of a trefoil knot with all negative crossings is 1 + x^2. I was therefore wondering, would the Conway polynomial of an equivalent trefoil knot with all positive ...
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Witten-Reshetikhin-Turaev (WRT) Invariant for surgery on the figure 8 at all roots of unity?

I've been trying to find a reference that gives the WRT invariant of a 3-manifold obtained by surgery on the figure 8 knot at an arbitrary root of unity but have only found them at the standard ...
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2answers
85 views

Why is the fundamental group of the Hopf link abelian but a two component unlink isn't

Title says it all basically. I'm trying to understand why the fundamental group of the hopf link (or really, the compliment of the Hopf link) is abelian. I mean, in a certain way I understand it, but ...
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2answers
64 views

Number of minima in a ribbon disk?

I am asking this question mainly with the hope of finding a reference to (presumably well-trodden) topic. Let $K$ be a ribbon knot and define $I(K)$ to be the minimum over number of minima of all ...
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1answer
38 views

Are link (non-splittable) quandles complete invariant up to orientation?

I am reading knot quandles. I read that knot quandles are complete knot invariant upto orientation from thesis of David joyce, An Algebraic Approach to Symmetry with Applications to Knot Theory, page ...
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Kauffman bracket for oriented links

When defining the Kauffman bracket, the skein relation is usually given for unoriented links. But if I put some orientation, one of the resolutions could potentially become contradictory. As an ...
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32 views

show that Jones two-variable polynomial can be expressed through finite type invariant. [duplicate]

I have this questions that says: prove that the two-variable Jones polynomial can be expressed through Finite type invariant. can somebody please explain how can this be done? thank you so much in ...
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Computing Incorrect Signs for Planar Graph of an Alternating Knot Projection

I'm trying to work through the The Knot Book by Colin Adams, and I seem to be consistently coming up with edges for a planar graph of the figure-eight knot where there are both positive and negative ...
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1answer
34 views

Khovanov homology over vector spaces, Z-modules or groups?

When I read various papers on Khovanov homology, sometimes it is defined in terms of graded vector spaces, sometimes as graded $\mathbb{Z}$-modules. Is there a difference? E.g. can the vector field ...
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A question about Witten's understanding of Jones polynomial

In his landmark paper, Witten proposed that the Jones polynomial can be obtained by the expectation value of the Wilson loop operators over links in the Chern Simons action. He wrote that one can ...
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1answer
56 views

Spanning trees of the Tait graph and Khovanov homology.

I'm reading this paper by Champanerkar and Kofman which proposes a nice way to compute the Khovanov homology of a link (diagram) via spaning trees of its Tait graph. The authors actualy define a ...
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1answer
53 views

Degree shift operation when constructing chain complexes of the Khovanov Homology

I'm reading this paper by Dror Bar-Natan, On Khovanov’s categorification of the Jones polynomial (here). On chapter 3 (categorification), he wrote: With every vertex $\alpha\in \{0,1 \}^\chi$ of the ...
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1answer
78 views

Khovanov Homology

I'm reading this and trying to understand how he computed the Khovanov homology of the Hopf link. The construction of the chain complexes and the maps look fine to me but the only problem is, I do ...
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Knot invariant in arbitrary 3-manifold

In famous Witten's paper "Quantum Field Theory and Jones Polynomial", Witten proposed $\int DA \exp{iL} \prod_{k=1}^{r} W_{R_t} (C_i)$ as the knot invariant in ARBITRARY 3-manifold. ($L$ is a ...
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Quantum Groups for Generic q and 3d-TQFT. What breaks?

I've just started looking through Quantum Invariants of Knots and 3-Manifolds by V.G Turaev and want to understand what exactly is breaking in the construction of a 3d-TQFT when one considers the ...
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1answer
188 views

Existence of transverse homotopy between knots in a 3-manifold

I have a 3-manifold $\Sigma$ and two homotopic embedded knots $K_{0}(t): S^{1} \to \Sigma$ and $K_{1}(t): S^{1} \to \Sigma$. I wish to refine the homotopy between them to a "transverse homotopy" i.e, ...
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What are the Jones polynomials for the torus links and the closure of the other braid word below?

I am working on a project to determine the Jones polynomial for the torus links and a class of links which I call tst links. Their braid words are respectively given by $$(\sigma_1 \sigma_2 \cdots \...
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2answers
129 views

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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1answer
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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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4answers
265 views

Proving a sharp maximum identity for some relatively prime odd integers

Let $p, q$ and $r$ be relatively prime odd positive integers satisfying $$pq + pr - qr = 1$$ and $$1<p<q<r.$$ Define $$f(a,n)= -(q + r)a^2 + 4aqn - 4(q - p)n^2 - 4n$$ where $-p \leq a \leq p+...
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1answer
34 views

How does the knot orientation interact with other properties or invariants.

The knot table consists of prime, non-oriented knots, not including the mirror images. However, for signature to be defined, we have to assign orientation to knots. How does the orientation influence ...
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1answer
47 views

Seifert Matrix of Amphichiral Knots

In Murasugi "Knot Theory and Its Applications" 5.4.7 I found an information that given the knot $K$ with the Seifert matrix $M_K$, the Seifert matrix $M_{K*}$ of its mirror image $K*$ is $S$-...
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0answers
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Invariant of free knots

In section 5. of https://arxiv.org/pdf/0912.5348.pdf, prof. Manturov gives a definition of an invariant of free knots, the $[\cdot]$ invariant. It is considered as an element of the space $\mathbb{Z}...
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1answer
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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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1answer
93 views

Quandle homomorphism does not always induces group homomorphim on inner automorphism group of quandles.

Let $X$ and $Y$ be two quandles and $f: X \rightarrow Y$ be a quandle homomorphism. Then we can define a map $\bar f: Inn(X) \rightarrow Inn(Y)$ as $\bar f(S_a)=S_{f(a)}$, where $a \in X$. Then $\bar ...
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1answer
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Multivariate Alexander polynomial vs single variable Alexander polynomial

I consider the multivariate Alexander polynomial $\Delta(t_1,\ldots,L_n)$ for a $n$-component link (defined using e.g. the Fox derivative). If we wish to construct a 1-variable polynomial $A(t)$, we ...
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Presentation of $\mathbb{R}P^3$ in the Reshetikhin-Turaev Construction

In this video at 1:02:00 the speaker gives a presentation for $\mathbb{R}P^3$ while discussing the Reshetikhin-Turaev Construction. I'm struggling to figure out what that presentation has to do with $...
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1answer
55 views

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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1answer
60 views

Framed Links are Ribbon Graphs

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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1answer
67 views

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 ...
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1answer
88 views

Crossing types in Gauss codes

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 ...
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1answer
86 views

Prove the Homfly polynomial of a link is determined by its Skein relations

One of the questions from a past paper on my Knot Theory course is to prove that $P(L)$ (the Homfly polynomial of a link $L$) is determined by: $$ (1) \ \ \ P(U_1)=1\\ (2) \ \ \ q^{-1}P(L_+)-qP(L_-)=...
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1answer
52 views

Quantum degree of Khovanov Complex

Reading Turner's 5 Lectures on Khovanov Homology (here) , I am trying to understand how to do computations using the long exact sequence defined on Lecture $3$. I have realized that I cannot ...
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1answer
103 views

Classification of polynomial knot invariants?

I am trying to find some information regarding the differences between the following knot invariants: Conway, Jones, Kauffman and HOMFLY. I know that HOMFLY can be reduced to Conway and Jones. And I ...
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80 views

Unknotting number conjecture

In Knots and Links by Cromwell he states that is it conjectured that this upper bound of the unknotting number of a composite link is in fact an equality $$u(L_1\#L_2) \leq u(L_1) + u(L_2) $$ I am ...
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1answer
46 views

Software for computing virtual knot invariants

Do you know any software which computes invariants for a given virtual knot? I mean invariants such as: Jones Polynomial (or even Khovanov homology), etc.
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Is there a name for the family of knots beginning with $6_3$, $8_7$, and $10_5$?

In my research, I've been playing around with a matrix representation of knots from which various knot invariants are calculable, and I have noticed several families of knots which I don't believe ...
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how to discretize gauss linking number integral to use in polymer simulations with finite number of monomers?

I need to discretise Gauss's linking number formula, to be used for finding the the linking number of a given topological conformation of say a ring like polymer chain (mainly Hopf link) . The ...
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1answer
85 views

Kontsevich invariant of surface knot

Kontsevich defined knot invariants by iterated integrals in the first half of 1990s in M. Kontsevich, Vassiliev's knot invariants, Adv. Sov. Math., 16(2) (1993) 137-150. After that Le, Murakami and ...
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89 views

Is this rule for the bracket polynomial redundant?

The rules for the bracket polynomial are generally given as: [https://tex.stackexchange.com/questions/306004/drawing-the-rules-for-the-bracket-polynomial] However, in my university lecture notes, it ...
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1answer
120 views

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 ...
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1answer
143 views

On Alexander polynomial of a knot

The Alexander polynomial of a knot is of the form $$\Delta(t)=det(V^T-tV),$$ where $V$ is the Seifert matrix, see http://archive.lib.msu.edu/crcmath/math/math/a/a116.htm. What is geometric or some ...