Questions tagged [knot-invariants]
For properties of knots that remain unaffected by Reidemeister moves
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Does the determinant of a Knot bound the number of primes for which the mod p rank is nonzero?
Definition. If $V$ is a Seifert matrix for a $\operatorname{knot} K$, then the determinant of $K$, denoted $\operatorname{det}(K)$, is the absolute value of the determinant of the symmetrization of ...
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Why the bridge index of $8_{10}$ is 3?
This might be too elementary. I tried to deform the projection but couldn’t be able to find a projection of knot $8_{10}$ with 3 maximal overpasses. Is there any elementary reference on calculating ...
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Is there an ill-embedded ball in the 4-sphere?
In https://arxiv.org/pdf/2102.04391.pdf, there is an explanation of how one could theoretically use a pair of knots $K$ and $K'$ (one slice and the other not) with the same 0-surgery to generate a ...
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$8_{17}$ knot is non-invertible [closed]
I'm finding everywhere this fact about this being the simplest knot which is not equivallent to its reverse (same knot, other orientation), but I can't find any proofs out there and I also don't know ...
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Almost all knots are non invertible
K. Murasugi mentions in p.45 of his book "Knot Theory and Its Aplications" that almost all knots are non invertible, meaning that they are not equivallent to their reverses, where the ...
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The degree of the Alexander polynomial is at most twice the genus.
The genus $g(K)$ of a knot $K$ is the minimum possible (topological) genus $g(S) = \frac{2-\chi(S)-B}{2}$ of a Seifert surface $S$ for the knot $K$ (where $\chi(S)$ denotes the Euler characteristic of ...
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Are there examples of different knots with identical Jones polynomials and different Seifert Genus?
I'm wondering if its ever possible to find two non-isotopic knots which have identical jones polynomials but different seifert genus?
Attempting to google for this I found this example of non-isotopic ...
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Alexander Polynomial
Recently I learned the Alexander Polynomial of a knot and how to find the polynomial for a given knot. Now there are some questions arise,
I am trying to give some classification of hyperbolic knots, ...
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How to understand the framing of a knot?
I was told the framing of a knot is the linking number of the push-off. But I don't understand why the framing does not depend on the knot but only on the parallel copy.
How about a Legendrian knot? (...
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Does every knot have a diagram such that every arc is a bridge?
Since the number of crossings in a knot diagram is the same as the number of arcs (or edges), can you construct a diagram of a knot where each arc crosses over exactly one other arc? My first thought ...
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Jones polynomial in Bar-Natan’s paper “On Khovanov’s Categorification of the Jones polynomial”
I’m reading Bar-Natan’s paper “on Khovanov’s categorification of the Jones polynomial”, I had previously been reading Lickorish’ book to have a good understanding on the Jones polynomial before diving ...
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Determinant of Alexander Matrix for Torus Links
The core of the problem
Let $q,r\in\mathbb N$ be natural numbers with greatest common divisor $d$. Consider the $(q-1)\times(q-1)$-matrix
$$
B:=\begin{pmatrix}
-1\\
1&-1\\
&1&-1\\
&&...
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Acyclic complex in Khovanov homology
I am reading Dror Bar-Natan's paper Categorification.
In section 3.5.1 (page 9), "Invariance under R1", it is claimed "It is easy to check that $\mathcal{C}'$ is subcomplex of $\...
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Struggling to Show Alexander Polynomial is a Knot Invariant Using Skein Relations
For (one of) the books I am using to learn knot theory, the Alexander polynomial is defined by the skein relation, or the unknot has polynomial 1 and the relation $\Delta(L_+)-\Delta(L_-)+(t^{1/2}-t^{-...
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Wirtinger's presentation gives that the link group of two disjoint circles is $\mathbb{Z}^2$?
Given two disjoint circles $a$ and $b$ that are projected in a way so that there's a positive and negative $a$-over-$b$ crossing, Wirtinger's presentation gives that the generators $a$ and $b$ commute ...
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Jones polynomials of alternating knots
Fox's Trapezoidal Conjecture asserts that the coefficients of the Alexander polynomial of an alternating knot alternate and the sequence of their absolute values forms a trapezoidal shape.
The same is ...
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How to tie a tight knot around my cable?
I would like to tie my new connector to my cable tightly using the provided string.
This is what the untied knot and the the tied knot look like; unfortunately, the adaptor can still slip and slide ...
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Kaufman Bracket on Links vs Framed Links
The book "Quantum Invariants: A Study of Knots, 3-Manifolds and Their Sets" by T. Ohtsuki gives the following definitions:
A framed link is the image of an embedding of a disjoint union of ...
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How to recognize a knot, practically?
This question was inspired by the very first exercise in Thurston's Three Dimensional Geometry and Topology, where he gives a picture of a very tangled up loop and asks what manifold it depicts.
I ...
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Does the Vassiliev $v_2$ invariant satisfy a relation similar to the Arf invariant?
The Arf knot invariant $\operatorname{Arf}:\{{\rm knots}\}\to\Bbb Z/2\Bbb Z$ satisfies the relation
$$\operatorname{Arf}(K_+) + \operatorname{Arf}(K_-) \equiv \operatorname{lk}(L_1, L_2) \pmod{2}$$
...
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How does KnotInfo classify knots based on DT Code?
The site KnotInfo can classify many small knots up to mirroring given a Dowker–Thistlethwaite Code of the knot, and can even identify whether a knot is a connected sum of other small knots.
Being very ...
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Why should boundary maps be degree 0 in Khovanov Homology?
In Bar-Natan's On Khovanov's categorification of the Jones polynomial (https://arxiv.org/abs/math/0201043), the claim in section 3.2 when constructing the differential $d_\xi$ is that $d_\xi$ ought to ...
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Jones polynomial of a knot in terms of its Seifert matrix
It is well known that the Alexander polynomial of a knot can be written in terms of the Seifert matrix of the knot by a simple relationship $$\Delta(t)=\det(V^T-tV),$$ where $t$ is a formal variable ...
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Jones polynomial of the left-handed Trefoil knot - which crossing for skein relation L_0?
I tried computing the Jones polynomial for the left-handed trefoil knot, but ran into a bit of an issue with how I pick my crossings for the L_0 skein relation.
I decided to work with the lower L_+ ...
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Whitehead double of a non-trivial knot is non-trivial
How can one show that if a Whitehead double of a knot is trivial, then the original knot must have been trivial? Since the Alexander polynomial and the signature vanishes, and since it is not clear ...
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What kind of mathematical knot is the square knot?
What kind of knot is the square knot? I made an attempt to calculate the Connway notation of the square knot and it did not match with any of the prime knots. Either I did the calculation wrong or the ...
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Proving that $n$-component Brunnian link is nontrivial
I stumbled upon the attached image. It shows a way to construct an $n$-component Brunnian link for any $n\geq 3$. That is, this link is not trivial, but deleting any of its components makes the new ...
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When a 3-strand pretzel link is an unknot?
I want to find any (nontrivial) necessary (and also sufficient, if possible) condition to determine whether $(p,q,r)$-pretzel link is equivalent to the trivial knot, where $p$, $q$, $r$ are integers.
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Is a complete knot polynomial known?
Wikipedia states in its article on knot invariant that
Other examples are knot polynomials, such as the Jones polynomial, which are currently among the most useful invariants for distinguishing knots ...
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Smooth 4-genus not agreeing with topological 4-genus intuition
I am aware that there are many examples of knots which have different topological 4-genus than smooth 4-genus (11n_34 is the one with the least amount of crossings, it is topologically slice but has ...
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If L=L1#L2 is a positive link, are L1 and L2 also positive links? What about adequateness?
Lets say I have an oriented positive link L (i.e. an oriented link which admits a diagram with only positive crossings) such that L=L1#L2 for two links L1, L2.
Now my question is wether those two ...
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Which hyperbolic fibered knots have monodromy with a single singularity?
The figure eight-knot has pseudo-Anosov monodromy with no singularity. I have read that the (-2,3,7)-pretzel knot has pseudo-Anosov monodromy with a single 18-prong singularity on the boundary of the ...
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Could anyone introduce some books about Knot Theory and polynomial?
I've already learnt basic topology, and understood part of differental manifolds, the embedding, and part of algerbraic topology, the simplicit homology.
My boss advised me to finish a subject review ...
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Every matrix $A$ with $\text{det}(A-A^T)=1$ can be realized as Seifert matrix
Let $A\in\text{Mat}(2n\times2n;\mathbb{Z})$ be an integer matrix with $\text{det}(A-A^T)=1$. I got told that every such matrix can be realized as Seifert matrix of a Seifert surface $F$ of a knot $K$ ...
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Does the Gordian distance depend on the chosen link diagram?
Let $K,L:S^1\rightarrow S^3$ be two knots. The Gordian distance between $K$ and $L$ is defined to be the minimum number of crossing changes to convert a knot diagram of $K$ to a knot diagram of $L$. ...
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Non ambient isotopic knots with same Seifert matrix
I am looking for an example of two knots $K,J$ and Seifert surfaces $F,G$ of $K$ respectively $J$, such that for an appropriate basis both surfaces admit the same Seifert matrix, but $K$ and $J$ are ...
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Does this knot collinearity condition have a nontrivial solution?
Recall that a link is an embedding of some number of circles in space up to isotopy. (Think "a knot but maybe multiple pieces.") Three oriented links form a skein triple if they have ...
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Alexander polynomial and orientation
This should be a simple question, but for some reason I can't seem to find an answer anywhere. My question is: is the Alexander polynomial defined as an invariant of oriented knots or of unoriented ...
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Are two Seifert surfaces with the same Seifert matrix ambient isotopic?
Are two Seifert surfaces with the same Seifert matrix ambient isotopic? I assume not, but it would be really helpful to have a counter example.
Thanks in advance!
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Practical applications of knot concordance
Could someone mention some of the practical applications of knot concordance? I was not able to find any from Google Scholar. Thanks in advance.
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For every prime $p$, does there exist a knot $K$ such that $K$ is $p$-colourable?
Once there is a $p$-colourable knot $K$, $K\#K$ is also $p$-colourable and, iteratively, there are infinite many $p$-colourable knots. But I am not sure if there really is a $p$-colourable knot for ...
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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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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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