Questions tagged [knot-invariants]

For properties of knots that remain unaffected by Reidmaster moves

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26 views

Unusual skein relation in HOMFLY polynomial

If I take the HOMFLY(PT) polynomial defined by $$l \,P(L_+) + l^{-1}\,P(L_-) + m\,P(L_0) = 0,$$ I have looked at expressions of the form (knots that are the same except inside a small disk, where ...
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50 views

Given a sequence of head to tail vectors forming a closed loop, how can I determine if they form a knot? [closed]

Consider if we have some sequence of vectors placed head to tail which form a closed loop. How can one determine whether they form a loop? We assume that it is given that the vectors close, that is ...
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38 views

A knot $K = K_1 \# K_2$ is alternating if and only if $K_1$ and $K_2$ are alternating.

Is a knot $K = K_1 \# K_2$ is alternating if and only if $K_1$ and $K_2$ are alternating? In particular, I'm interested in the following direction: If $K_1 + K_2$ is an alternating knot, are both $...
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14 views

Does the link with trivial knot group trivial? [duplicate]

I know that if a the knot group of a classical knot is isomorphic to the infinite cyclic group, then the knot is unknotted. How about the link, is this result also valid for links. In other words, if ...
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1answer
32 views

Classification of knots with unknotting number 1.

Is there a classification of knots with unknotting number=1? I can think of one infinite family at least. That is, twist the unknot $n$ times then join the two ends together. In fact, I think maybe ...
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5 views

Does a succession of knot slicings create a 4 dimensional knot?

Consider the unknot floating about and then it curls up and a line passes through another line and it becomes a trefoil knot. Then a line passes through a line again at it becomes the unknot. If we ...
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64 views

Do there exist non-trivial knots whose Jones polynomial is a unit?

Question: It's an open problem whether or not the Jones polynomial distinguishes the unknot from all other knots. That is, the following problem is unsolved. Does there exist a knot $K$ which is ...
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1answer
35 views

Is the linking number via Seifert surfaces well defined?

Let $i(K, F_L)$ be the signed count of intersections of an oriented knot $K$ with a Seifert surface $F_L$. (That is, $F_L$ is an oriented compact surface with boundary $L$ for some knot $L$.) I want ...
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1answer
113 views

A Tensor Calculation with Braids

I am trying to follow the derivation of the Jone's Polynomial from a braid representation presented in chapter 2 of Ohtsuki's Quantum Invariants. The representation of the braid $b$ with $n$ strands ...
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31 views

computing the Kauffman bracket with the given relation

My Problem: Use the relation to compute the bracket of the link diagram $D_n$ with $n$ components: My attempt: It seems to me that raising the given equation to the $n^{th}$ power is the most ...
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39 views

Algebraically slice implies slice

I am studying an article by Livingston ("New examples of non-slice, algebraically slice knots", Proceeding of the AMS, 2001) about an example of an infinite class of knots which are algebraically ...
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1answer
47 views

Alexander polynomial of any knot evaluated at 1 is $\pm$ 1

I'm supposed to prove for a knot theory homework assignment that the Alexander polynomial of any knot (as opposed to link) is $\pm1$. From examples, I'm pretty convinced that this is true, but I have ...
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Equivalence of two elements in $\pi_2(BX)$, where $BX$ is the rack space corresponding to the quandle $X$.

I am reading the proof of Lemma 4.1 given in the chapter "Some of Quandle Cocycle Invariants of links" of the book "Quandles and Topological Pairs" by "Nosaka." Before coming to the question, I have ...
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1answer
38 views

Reference request: Knots that don't come from Milnor spheres.

In Milnor's book "Hypersurface singularities" He discusses shortly knots that arrive as Milnor spheres of algebraic curves, i.e knots that are the intersection of a $3$ sphere around a singular point ...
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17 views

Ribbon picture of single strand with a knot

I was reading Steven Simon's notes on Topology to understand few ideas used in condensed matter theory. I want to declare that I don't have any formal knowledge of knots and topology in general. So ...
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1answer
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Maximize integral over a 3D knot - how to make my intuition rigorous?

Yesterday, I was thinking about creating a function that distinguishes between different topological knots (embeddings of $S^1$ into $\mathbb R^3$) and I came up with the following function $f$. If $K$...
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23 views

Question in knot theory , the relation between the Jones polynomial and the Homfly polynomial

How can I show that the HOMFLY polynomial determines the Jones polynomial of a link. I'll highly appreciate help.
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Difference between framing and writhe of a knot

The writhe of an oriented knot is the number of positive crossings minus the number of negative crossings while the framing of a knot is defined to be the linking number of the knot with the curve ...
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1answer
23 views

Reference for basic examples of skein algebras

I’m aware of the magic of skein algebras of surfaces only recently. It connects to knot theory in several ways. But while there’s abundance of basic texts on knot theory, I cannot find some for skein ...
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89 views

Knot group is $\mathbb{Z}$ iff $K$ is the unknot

Let $K \subset S^3$ be a knot; we call knot group the fundamental group of the complementary of $K$ in $S^3$. I've come across the fact that the only knot whose knot group is isomorphic to $\mathbb{Z}...
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3answers
69 views

Computation of the A-polynomial of the trefoil knot

What is the A-polynomial of the trefoil knot? Show the steps. I would like to compute the A-polynomial of the trefoil knot, whose fundamental group is given by: $$\langle x, y | x^2 = y^3 \rangle$$ ...
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A factorization result for links

I am currently informing myself about the basics of knot and link theory. I learnt pretty quickly about prime knots and a unique decomposition theorem for knots stating : $\text{Thm: Every (isotopy ...
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1answer
38 views

Classification of links with unknot components

Question I am interested in the links in $\mathbb{R}^3$ with trivial components. More precisely, I'd like to know if the classification of links with finitely many components, which are all unknots ...
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35 views

A question in knots

Prove that if the orientation on one component of a two-component oriented link L is reversed then its linking number is negated. What is the linking number of the mirror-image link $\bar{L}$ ? Would ...
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Skein relations and jones polynomial

I apologize for my idiot question but it just is not clear to me how skein relations with coefficient , A, of higher powers is useful? For higher powers of A = x^-1/2 - x^1/2 we have something like: ...
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Does any proof of the knottedness of the trefoil capture our intuition?

The fact that trefoil knot is knotted, i.e. not equivalent to the trivial knot, was first proved in the early 20th century by Wirtinger and Tietze. They used knot groups, though there are now ...
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1answer
128 views

How do you prove Reidemeister’s theorem for wild knots?

Reidemeister’s theorem states that any two diagrams for the same knot are related by the three Reidemeister moves. Now the way Reidemeister, Alexander, and Briggs originally proved it is by showing ...
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1answer
52 views

Alexander polynomial evaluated at -1 and p-colorability

I have been working with a group studying generalizations of Fox's $p$-colorability / $p$-labelability and I've seen stated in several places that a knot is $p$-colorable if and only if the Alexander ...
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1answer
90 views

Can a two-bridge torus knot be slice?

For a 2-bridge torus knot of the form $T(2,r^2)$, where $r^2$ is an odd number, we have that its determinant is $r^2$, that is a condition that we need for a knot to be slice. Also its Alexander ...
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Homology of $X_{\infty}$ space for a given Seifert surface of an oriented link $L$.

I am reading the proof of Theorem 6.5 given in the chapter "The Alexander Polynomial" of the book "An introduction to knot theory" by "Lickorish". I have some doubts related to the homology of spaces ...
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42 views

Resource for double branched covers

I am looking for a resource to learn about the double branched cover of a knot. How do we construct them? Given a knot in $S^3$, is there a way to get a cell decomposition of the double branched cover?...
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95 views

Why the Alexander polynomial of the unknot (trivial knot ) is the constant polynomial $1?$

The book said this: Why the Alexander polynomial of the unknot (trivial knot ) is the constant polynomial $1?$ could anyone explain this for me please? And is the subscript $k$ like the ...
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1answer
67 views

Determinant of rational links

On page 9 of his book "An Introduction to Knot Theory", Lickorish just mentions the fact that the determinant of the rational knot $K(p,q)$ is $|p|$. I'm trying to find a proof of this fact (I ...
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1answer
53 views

How the components of alternating link diagram $D$ are boundaries of the regions of one color after performing positive smoothing at all crossings?

I am reading the proof of the Proposition 5.3. of the chapter "The Jones Polynomial of an Alternating Link" from the book "Introduction to knot theory" by "Lickorish". I have a problem with ...
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1answer
52 views

If L is a split link then L has split link diagram. Is this true?

I am reading the book "An introduction to knot theory" by Lickorish. Following are the definitions at the beginning of the chapter "Geometry of Alternating Links". A link $L$ in $ \mathbb{S}^3$ ...
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1answer
111 views

What is the Alexander polynomial of knot $6_{1}$?

My Alexander polynomial for $6_{1}$ knot is $t^4−5t^3+8t^2−5t+1$ , so its span is 4, so the lower bound of its genus is (1/2 span of its Alexander polynomial) so the lower bound of the genus is equal ...
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1answer
139 views

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 ...
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Knotoids in 3D: Constraints on endpoint lines?

One attractive model for Turaev's knotoids1 is illustrated below: A curve connects two points confined to vertical lines—call them $L_1$ and $L_2$— in $\mathbb{R}^3$: Gügümcü, Neslihan,...
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71 views

Derivation of Jones-polynomial from HOMFLY-polynomial

I came across something seemingly trivial, but I don't know why this mistake happens. We have the HOMFLY-polynomial $P(L)\in\mathbb{Z}[l^{\pm1},m^{\pm1}]$ for oriented links $L$, which satisfies: 1) ...
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1answer
108 views

Algebraic equations for knots

Is there any general method to find out algebraic equations rather than parametric equations for a knot? That is, I want a set of two equations in $\mathbb{R}^3$ whose common solution is the given ...
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1answer
113 views

Left- and right-handed trefoil knot and its mirror image.

I know that trefoil knots do not have a mirror image, but I also know that there is a left-handed and a right-handed trefoil knot. I have the drawings of them. Isn't this a contradiction?
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27 views

Show that if K is any knot in regular position there is an alternating knot (in regular position) that has the same projection as K.

Show that if K is any knot in regular position there is an alternating knot (in regular position) that has the same projection as K. Could anyone give me a hint for solving this please?
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42 views

Showing that amphicheirality of 2 knots implies their equivalence.

The question is: If $K_{1}$,$K_{2}$ are amphicherial knots. Show that $K_{1} \cong K_{2}$. The definition of amphicheiral is as follows: A knot $K$ is amphicheiral iff there exists an orientation ...
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35 views

The relation between mirror image and equivalence of two knots.

The question is: Let $T(x, y, z) = (x, y , z)$ and $K_{1}$,$K_{2}$ are 2 knots such that $T(K_{1}) = K_{2}$. Show that $K_{1} \cong K_{2}$. Where $K_{1} \cong K_{2}$ is defined below: The ...
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Showing that two knots are equivalent if an invertible linear transformation maps one onto another.

The question is: Let $K_{1}$,$K_{2}$ be two knots in $\mathbb{R^3}$. Let $T: \mathbb{R^3} \rightarrow \mathbb{R^3}$ be an invertible linear transformation such that $T(K_{1}) = K_{2}$. Show that $K_{...
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2answers
61 views

Prove that the equivalence of 2 knots is an equivalence relation.

The definition of equivalence of 2 knots according to Richard H. Crowell and Ralph H. Fox, edition 1963, is: Assume that $K_{1}$,$K_{2}$ are 2 knots in $\mathbb{R^3}$, then they are equivalent , ...
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35 views

Show that the number of tame knot types is at most countable.

Show that the number of tame knot types is at most countable. I want a hint for solving this problem please.
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20 views

The regions into which $\mathbb{R^2}$ is divided by regular projections can be colored into white and black.

Show that the regions into which $\mathbb{R^2}$ is divided by a regular projection can be colored white and black in such a way that adjacent regions are of opposite colors ( as on chessbord). Could ...
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68 views

Devise a method for constructing a table of knots, and use it to find $10$ knots of not more than $6$ crossings.

Devise a method for constructing a table of knots, and use it to find $10$ knots of not more than $6$ crossings (do not consider the question of whether these are really distinct types.) Could anyone ...
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1answer
90 views

Show that there are no knotted quadrilaterals or pentagons. what knot types are represented by hexagons? by septagons?

I am reading the book of Richard H. Crowell of "Introduction to knot theory" And in chapter I, it has this question Show that there are no knotted quadrilaterals or pentagons. what knot types are ...

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