Questions tagged [knot-invariants]

For properties of knots that remain unaffected by Reidemeister moves

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Split link properties

I am stucked, to show the following statement: Let $L = L_1 \cup L_2$ be a split link. Show that $det(L)=0$ but $\sigma(L)= \sigma(L_1) + \sigma (L_2)$. Where $det$ means the determinant of the link ...
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what are vassiliev invariants and how are they related to Chern-Simmons?

Possibly this question is very open for this platform and a little diffuse but I am a physicist and my knowledge in tides is not that high but I do not understand graphically what the vassiliev ...
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Trying to explain the unknotting number of $7_5$

I am trying to proof that the unknotting number of prime knot $7_5$ is $2$. For this purpose, I am studying the minimal number of crossings that need to be changed in the knot in order to get a ...
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how did they come up with polynomials as knot invariant

I understand how to calculate a Jones polynomial for a given knot, but I am not sure why would one search for a polynomial for invariants. How did he come up with these calculation rules?
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Why is the determinant of the 5-2 knot not a multiple of 5?

A knot $K$ is $p$-colorable if and only if the det($K$) is divisible by $p$. So why is the |det($5_2$)| $=7$, and not a multiple of 5, especially when there are only 5 arcs in the knot?
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If two knots are tricolourable does this imply that they are 3 equivalent?

In reading the knot book there was a question regarding that 3 moves preserve tricolouration and pointing to if a knot is tricolourable and 3 equivalent to another knot then the other knot is also ...
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Why is Conway notation relevant, instead of braid notation (for knots)

In the early XXth century, Alexander showed that every knot is the closure of a braid, and Markov gave necessary and sufficient conditions for two braids to have the same closure. My question is: why ...
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Non-orientable genus of Unknot and it's uniqueness

In the book "Topology Now!" by Robert Messer one of the practice problem suggests, " One could define the nonorientable genus of a knot to be zero for the trivial knot, and for any ...
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Explicit form of element in a link group

I have a link which is union of knots $K_1\cup\ldots\cup K_n.$ I do know how to find link group $\pi_1(\mathbb{R^3}-K_1\cup\ldots\cup K_{n-1})$, for example, using Wirtinger presentation. What I want ...
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Genus zero links

We know that every link in $S^3$ bounds a Seifert surface, and the genus of a link is defined as the minimal genus among the Seifert surfaces it bounds. It is easy to see that the only genus zero knot ...
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Why can Rolfsen use Van Kampen? (Knots and Links)

I am having some difficulty with Rolfsen's derivation of the Wirtinger presentation of a knot in "Knots and Links" (pages 56 to 60). The basic setup is illustrated below. The proof begins ...
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How to prove that braid index of a specific knot is at least 3?

I have a specific knot $K$ (it seems that it is $6_3$) and want to find its braid index. I managed to construct a braid with 3 strings whose closure is $K$, however I do not know whether 2 strings ...
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What is the bilinear intersection form of real homology of 4-manifold?

I am currently reading Invariants of 3-manifolds via link polynomials and quantum groups by N. Reshetikhin and V.G. Turaev. In section 3.2, given a framed link $L$, we can get a $4$-manifold $D_L$ by ...
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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 (...
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Good invariant quantifying how complex a knot is

I am currently trying to get upper bounds on the unknotting numbers of a list of knots. What I did is the following: I start with a diagram of the knot, then for each crossing, I simplify the knot ...
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Relationship between links

How do I find relationships between $$ lk(L),\; lk(mL),\; lk(rL),\; lk\Big(m \big(r(L)\big)\Big),$$ for any arbitrary oriented link $L$, where $m$ denotes the mirror and $r$ denotes the reverse. So ...
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HOMFLY polynomial is invariant under Reidemeister moves

Is there an easy way to verify that the HOMFLY polynomial is invariant under Reidemeister moves? For Jones polynomial, it is obvious by the way it is defined (not via skein relations). It is proved in ...
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How to recognize torus knots

I have a list of 185 knots with less than 15 crossings and I want to check which of them are torus knots. I know there are many invariants being able to exclude some of them being torus, but I was ...
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Singular framing of a singular knot

In "Introduction to Vassiliev Knot Invariants" (https://arxiv.org/abs/1103.5628), singular framing is defined in p.82, and I think that we can twist the framing arbitrarily around zero ...
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Definition of Vassiliev invariants for framed oriented knots

In Bar-Natan's paper "On the Vassiliev Knot Invariants", he says that it is not difficult to extend the definition of Vassiliev invariants for (unframed) knots to framed knots, but I don't ...
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How can I proof: Figure-Eight Knot is 5-colorable?

I am learning Knot Theory recently. So I am doing some hand calculation for to find p-color for a Knot. So my current understanding says that A given Knot can be p-colorable if and only if the Knot ...
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What do variables $A$ and $t$ represent in the normalized bracket polynomial and jones polynomial respectively?

I am putting together a presentation on the Jones polynomial and some of its applications. I have been using The Knot Book by Colin C. Adams as my primary reference and have been supplementing it ...
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Is there an example of two topologically different knots with the exact same Conway notation?

on whether a Conway notation describes a unique knot, so it'd be two knots which are fundamentally topologically different
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Existence of Dowker Notation

If a knot diagram is oriented, and we follow this orientation, labelling each crossing with consecutive integers, then each crossing will be numbered with an even and an odd number. From this ...
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Skein Relations for Alexander Polynomial of Figure 8 knot - Example not working out

I'm trying to use the relationship between the Skein relations and the Alexander polynomial of a knot: $$ \Delta_{L+}(t) = \Delta_{L-}(t) + (t^{1/2} - t^{-1/2})\Delta_{L_0}(t) $$ to find the Alexander ...
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Longitude of knot

I am trying to understand the peripheral system of knot. In that direction, I don't see why the description of longitude of knot in Remark 3.13 does actually lie second commutator as shown in ...
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Number of ways n unknots can be configured into an n-bridge link?

I have n-unknots (trivial knots) embedded in $S^3$ with the condition that they form an n-bridge link. I'm trying to find the different ways they can be put together. So far I only have the trivial ...
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Finding the minimal crossing number for a knot

Can we find the minimal crossing number for a given knot or prove that a given diagram or code for a knot is the one with the minimal number of crossings? For the 8-knot it always has at least four ...
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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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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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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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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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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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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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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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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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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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