Questions tagged [knot-invariants]

For properties of knots that remain unaffected by Reidemeister moves

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Image of elements of $S_n$ in the Temperley-Lieb algebra

Consider the algebra $A_n$ generated by $u_1,\ldots,u_{n-1}$ subject to relations $u_i^2=-2u_i$, $u_iu_{j}u_i=u_i$ for $|i-j|=1$ and $u_iu_j=u_ju_i$ for $|i-j| \geqslant 2$. The algebra $A_n$ is the ...
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67 views

What is a slice knot?

so I've recently come across Lisa Piccirillo's proof on the Conway Knot Problem, and I want to learn more about it, but I can't really understand what is meant by whether or not a knot is "slice&...
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Computation of Colored HOMFLY Polynomials

I am trying to understand the colored HOMFLY polynomials. The theoretic description Anna Aiston gave in her PhD thesis is really nice, but what about the computation? I would like to understand the ...
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Application of higher-dimensional knot theory

There are quite a few applications to classical (1-dimensional) knot theory, such as knotted proteins, DNA,... Are there any potential applications of 2-dimensional knot theory? E.g. knotted surfaces ...
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Analogy to Van kampen theorem for Quandles

This is from thesis of David joyce, An Algebraic Approach to Symmetry with Applications to Knot Theory Please help me to understand the $\lim$ $AQ(U_{i}$,$U_{i} \cap K) $as given on page 42 of the ...
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Knots with infinite crossing

For me knots are embedding of $S^1$ in $\mathbb{R}^3$. I have following questions: Will knots have infinite crossing? If so, Why are we considering only knots with finite crossing ? Can someone ...
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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 ...
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Questions on the symmetry of of the alexander polynomial and the rank of the Seifert matrix

For my bachelor thesis I am using the book "Lectures on the topology of 3-manifolds. an introduction to the Casson invariant"(1999) by Nikolai Saveliev. Regarding the Alexander Polynomial as ...
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Linking number of torus links

Let $(p,q)$ be a pair of coprime integers; the torus knot $K(p,q)$ is the unique (up to isotopy) curve on the boundary of a solid torus which is homologous to $p\cdot \lambda + q \cdot \mu$, where $\...
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Can a diagram of knot with minimal number of crossings contains a degenerate crossing or cancelling pair of crossings?

Given a knot diagram $D$ and a quandle $Q$. One can associate an element of $Q$ to an arc $\alpha$ of $D$, which called the color of $\alpha$ and denoted by $c(\alpha)$. By giving a color to every arc ...
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Does the factorization operation for composite knots applicable for any projection of the knot?

Let $K$ be a knot in 3-space. We know that if $K$ is not prime (composite) then $K$ can be represented as a connected sum of two knots, both are non-trivial. My question is : does this apply to any ...
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Prerequisites for Neuwirth's 'Knot Groups'

I have never studied knot theory before. I would like to get into the subject. I am interested in studying knots from a topological perspective (as opposed to a combinatorial one.) I am studying knot ...
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Two-torus embeddable knots

I was wondering if there were any known resources containing information on two-torus embeddable knots? I have checked the knot atlas with no luck, and it seems that many other cites only reference ...
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Using the Jones polynomial, determine which of the following links are equivalent or inequivalent?

Struggling to start this as not sure if to use skien relation or work out the writhe and kauffman brackets
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What would be Conway and Rolfsen style notations for four-component Brunnian link?

What would be Conway and Rolfsen style notation for the four-component Brunnian link given below: It was also asked about the link here: an old related question
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Mutant knots have isomorphic Skein Trees

I'm learning about skein relations and the Alexander Polynomial. I'm struggling understanding why mutant knots always have isomorphic skein trees. Does it have something to due with the fact that all ...
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Introduction to Colored Polynomial Invariants

I recently studied knot theory and made myself familiar with some knot invariants (e.g. Jones and HOMFLY polynomial). It was easy to find good books on that subject. Now I want to go further and learn ...
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Algorithm for tricoloring a knot

I have just started studying knot theory, and read about tricolorability of knots. Two knots are equivalent iff they are tricolorable. While trying to color a variety of complex knots, I tried a ...
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Does a classical link which admit the trivial coloring only for any quandle exist?

Let $L = L_1 \cup L_2 \cup \dots \cup L_n$ be a classical link of $n$ components. Suppose any coloring of $L$ by any quandle is the trivial coloring. That means a unique element in the quandle given ...
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Diagonalization over $\mathbb{Z}$

I'm studying knot theory and the textbook mentions that to find the mod $p$ rank of a knot, we take a certain matrix $M$ with integer entries corresponding to the knot and diagonalize it over $\mathbb{...
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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, ...
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Is it true that the generators of the link quandle corresponding to arcs in a reduced diagram of a link are pairwise distinct?

A reduced diagram of a link $L$ is a diagram $D$ without a nugatory crossing (isthmi) and it is such that the number of crossings cannot be reduced using a finite sequence of Reidemeister moves that ...
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Looking for formulas of colored jones polynomial

I have found equations for the colored jones polynomial of the trefoil knot as a function of $n$, where $n$ is the coloring. I have also found equations for the colored jones polynomial of $(p,m)$ ...
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What is the point of the differing appearances of the HOMFLY skein?

What is the point of the differing appearances of the HOMFLY skein? For example, can somebody explain to me why one may prefer the relation $tP_P(q,t)-t^{-1}P_-(q,t)=(q-q^{-1})P_0(q,t)$ to the ...
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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 ...
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Pronunciation of HOMFLY polynomial

How should HOMFLY polynomial be pronounced? It is an acronym, but clearly chosen so that it could be pronounced as a word.
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Definition of equivalence of knots

I've just started reading An Introduction to Knot Theory from W. B. Raymond Lickorish and I've come up with the following definition: Definition. Two knots $K_1$ and $K_2$ are equivalent if there ...
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151 views

Fixed points of strong inversion of a knot

(Edit: added definition of strong inversion.) (2nd edit: added motivation for question at the end.) Definition: A knot $K$ in $S^3$ is strongly invertible if there is an involution of $(S^3,K)$ which ...
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Is the surface bounding the complement of an unlink in 3-space incompressible?

Let $L$ be an unlink of $n>1$ components in $\mathbb{S}^3$. Let $N=\mathbb{S}^3-L$. Is the boundary of $N$ incompressible?
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Can we apply Reidemeister moves to self crossings of one component only of a “trivial” link and keep the others unchanged?

Suppose $L=L_1 \cup L_2 \cup L_3$ be a classical link of three components. Suppose $L$ is an unlink, that is $L$ can be splitted into three simple closed curves. Assume that $L$ has a diagram in 2-...
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Is there a way to get from a polynomial to a link with the polynomial as its Jones polynomial?

Sorry for the convoluted wording of my question. Suppose that I have some polynomial $P$. Is there an algorithm for getting a link $L$ such that $V(L)=P$? Is there an algorithm for getting the ...
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How can one find a knot with a given Alexander polynomial?

Suppose that I'm given a polynomial $P$ such that $P$ is the Alexander polynomial of some knot. Is there an algorithm to come up with a projection of a knot $K$ such that $\Delta(K)=P$? Are there ...
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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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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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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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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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112 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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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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155 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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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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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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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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102 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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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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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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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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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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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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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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