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Knot theory is dealing with embedding of curves in manifolds of dimension 3. A knot is a single circle embedded in the affine space of dimension 3 as a smooth curve not crossing itself. Many knot invariants are known and can be used to distinguish knots.

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Is there a combinatorial way to determine the coefficients of the universal finite-type invariant on a given knot?

There are various (equivalent?) descriptions of a universal finite-type knot invariant, e.g. https://arxiv.org/abs/q-alg/9603010. They take the form of formal power series valued in Feynman diagrams (...
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Notation for algebraic tangles

The Conway notation for rational knots is well known. I could use one for the more general algebraic tangles. As the name already says, one could build one out of tangle addition $+$, multiplication $*...
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The Kirby diagram of a manifold glued along the lens space $L(p,1)$

Suppose $K$ is a knot in $S^3$ with any framing and $m=m_0$ is its meridian with $-1$ framing. Suppose $m_1,\dots,m_{p-1}$ are unknots with framings $-2$, such that $m_{i-1}$ and $m_i$ are linked as a ...
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From braid representations to link invariants

If one has a $\mathbb{C}$-linear representation of the braid algebra into e.g. the Temperley-Lieb algebra i.e. $\rho:\mathbb{C}[B_{n}]\to TL_{n}(\delta)$, we can deduce a skein relation $\mathcal{S}$....
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Which knots appear as the singular locus of a polyhedral metric on the 3-sphere?

What can be said about a knot $K\subseteq S^3$ for which there exists a (Euclidean) polyhedral metric (aka Euclidean cone-manifold structure) on $S^3$ whose singular locus is precisely $K$? I'm ...
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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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93 views

Rack cohomology as derived functor cohomology

Let $X$ be a rack and $A$ be an $X$-module. By this paper, p. 33, we can associate a cochain complex $C^\bullet(X,A)$ to the pair $(X,A)$. This complex is explicitly defined by a differential $d$. I ...
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128 views

Braids of fibered knots

There are some theorems saying that the closure of a braid of a specific form is fibered. For instance, a theorem of Stallings says that the closure of a homogeneous braid is a fibered knot. I am ...
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113 views

Thompson's group F and algebraic links

There is a procedure, suggested by Vaughan Jones, which associates a link to every element of Thompson's group F. Also every knot or link in $\mathbb{R}^3$ can be obtained in this way. A subclass of ...
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171 views

Decidability of knot equivalence in general 3-manifolds? Surface equivalence?

Given a closed orientable 3-manifold $M^3$ and two knots $K_1$ and $K_2$ in $M$, is there an algorithm to decide if $K_1$ and $K_2$ are isotopic? Is there an algorithm to decide if there is a ...
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99 views

What is the determinant of the R-matrix defining the colored Jones polynomial?

Let $V_n$ be the $(n+1)$-dimensional irreducible representation of $\mathcal U = \mathcal{U}_q(\mathfrak{sl}_2)$, and let $\mathbf R \in \mathcal{U} \widehat \otimes \mathcal{U}$ be the universal $R$-...
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Questions on the proof Lemma 4.5 GTM 175, Lickorish

I am reading GTM 175 An introduction to knot theory by Lickorish and have some questions on the proof of Lemma 4.5 given. For (a), it says "Suppose that $C$ is amongst the $n$ components of $F\...
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101 views

Rolling wheel unicycle knots

Let $K$ be a knot, and $K(t)$ a parametrization of a space curve that realizes $K$. Roll a wheel $W$ of radius $r$ on $K(t)$ so that $W$ remains in the tangent-normal plane. Now track the wheel's ...
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208 views

Are knot invariants topological invariants? [closed]

I am a bit confused about terminology considering topology and knot theory. A topological invariant is considered to be a topological property that does not change under a homeomorphism of the space. ...
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Skein relation, Braids, and Hecke algebra

Many knot invariants (e.g. Alexander polynomial, Jones Polynomial,etc) admit a recursive algorithm based on the so-called skein relation But why the skein relation is a natural thing? People have been ...
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142 views

Questions about a few terminologies in “Knots and Links” by Rolfsen

In "Knots and Links" by Rolfsen, he mentioned words like *"the collar of a boundary", "bicollared boundary", "a bicollar on the boundary". I just wonder what ...
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Distinguishable knots (with constraints) over polyhedra

I'm trying to find the number of distinguishable knot projections over certain convex regular polyhedra according to the following constraints. On each face on the polyhedron the knot will have a ...
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1answer
136 views

Surveys on unknotting number

Any knot diagram could be converted to an unknot by cross change. The unknotting number of a knot diagram is the minimal number of cross changes needed. A knot could have many different diagrams and ...
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Can the Chern-Simons invariant of a cusped hyperbolic $3$-manfiold be defined mod $\mathbb Z$?

For a closed hyperbolic $3$-manifold $M$, the Chern-Simons invariant $CS(M)$ can be defined as an element of $\mathbb R/\mathbb Z$. When $M$ is cusped it can still be defined, but is now only defined ...
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1answer
60 views

Knots with a braid presentation with only positive or negative crossings on each fixed position

I am interested in the following class of knots $K$: $\{$$K$ has a braid presentation such that for any fixed position $k$, either only positive or negative powers of $\sigma_k$ appear in the braid ...
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630 views

Several questions about Gauss's mathematical conception of braids

I'm trying to figure out several things about Gauss's thoughts concerning a certain four-strand braid. The reference my questions are based on is mainly Moritz Epple's excellent article "orbits ...
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1answer
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Does a periodic and strongly invertible knot have a specific type of diagram?

Let $K \subset S^3$ be a knot which is both strongly invertible and periodic, that is, $K$ is fixed by both a smooth involution $\tau: S^3 \to S^3$ which preserves the orientation of $S^3$ but ...
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597 views

Knot theory and creative writing

I am a Ph.D. Candidate in Creative Writing and an M.S. Student in Mathematics and I'm writing my master's thesis on knot theory and trying to tie in applications to creative writing. Has anyone come ...
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1answer
176 views

Invariants of virtual knots?

Which invariants of classical knots are known to extend to virtual ones? In literature I have only found the Alexander polynomial to be defined for virtual knots.
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Handle decompositions of slice and ribbon disk exteriors

Let $K$ be a slice knot in $S^3 = \partial B^4$. Then $K$ bounds a smoothly properly embedded disk $D$ in $B^4$. Let $\nu(D)$ denotes the tubular neighborhood of $D$. Or we consider ribbon disks by ...
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(Best) ways to reduce knot complexity?

Lets say I have a diagram of a knot in some notation. What's the fastest algorithm to simplify it? Or asked differently: what algorithms do software usually use? I do not need to put it into the very ...
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198 views

Mazur homology spheres

This is Example 6.47 in Saveliev's book Invariants for homology $3$-spheres: Let us consider a two-component link $\mathcal L = L_1 \cup L_2$ in $S^3$ such that $\mathrm{lk}(L_1,L_2) = \pm 1$ and the ...
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Two knots with same Dowker-Thistlethwaite code have isomorphic knot groups?

Given a (tame) knot diagram, one derives the Dowker-Thistlethwaite code by travelling around the knot and numbering each crossing 1,2,3,.... A negative sign is given to an even number if you cross on ...
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1answer
174 views

$0$-surgery of slice knots and contractible manifolds

We know that if we attach $4$-dimensional $2$-handle $D^2 \times D^2$ to $S^1 \times S^2$, then we produce a contractible $4$-manifold. In this case, $S^1 \times S^2$ is $0$-surgery on the unknot. If ...
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94 views

Actions of two types of Kauffman skein categories

Consider the quotient of the monoidal category of framed tangles by one of the two skein relations together with the twist and dimension relations Here $1_\mathbb{1}$ denotes the identity morphism ...
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1answer
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Exchanging the components of a two-component link

Given a 2-component link in $S^3$ whose components are trivial knots, is it always possible to find a homeomorphism of $S^3$ that exchanges the components? I guess the answer is "no" (but I ...
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172 views

Possible “binomial” formula for the Jones polynomial

The following conjectural "binomial" formula for the Jones polynomials $$J(q)=(-1)^{n_-}q^{n_+-2n_-}\left(\sum_{k=0}^N\binom{N}{k}(-q)^k (q+1/q)^{\ell_{k+1}-1}\right)$$ is for a knot or link ...
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2answers
229 views

Unknotting in Morse notation without introducing new strands

A knot can be represented with a Morse link presentation, as a combination of cups, caps and crossings (which is not uniquely determined by the knot, of course): Two Morse link presentations of the ...
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Would it be simpler, pedagogically speaking, if textbook writers introduced root systems as an example of a quandle?

I could never, for the life of me, recall the definition of a root system in Lie theory. It probably doesn't help that I've never taken a course on Lie Theory - the algebra, or the groups, or the ...
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Using a 4th dimension to make Seifert surfaces isotopic?

Let $L$ be a link in three manifold $M^3$ and let $F_1$ and $F_2$ be two homeomorphic surfaces in $M$ with $L = \partial F_1 = \partial F_2$. Suppose that $F_1$ and $F_2$ are not isotopic rel ...
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1answer
168 views

Singularities of PL embedding of surface in a contractible 4-manifold

I am trying to understand the article "A solution to a conjecture of Zeeman" by Akbulut, but I am not an expert in PL-geometry. As far as I understand, two statements should be true, but I ...
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566 views

Knots realized as algebraic curves

Two questions: Q1. Have researchers worked out minimum-degree real algebraic curves in $\mathbb{R}^3$ realizing specific knots? Some work on the trefoil is reported in this MSE question.   &...
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Assigning a “canonical geometry” to a Seifert surface

I originally posted this on stackexchange, but it hasn't gotten an answer. I hope it's not inappropriate for this forum. Suppose I have a knot $K: S^1 \hookrightarrow S^3$ with minimal genus Seifert ...
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Bipartedly slice links and their surgeries

A link L in $S^3$ is said to be strongly slice if $L=∂D$,where $D$ is a disjoint union of smoothly and properly embedded disks in $B^4$. A link $L$ in $S^3$ is called bipartedly slice if $L = L_1 \cup ...
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Maximal Thurston--Bennequin number of boundary knot classes in contact handlebodies

Let $H$ be a contact handlebody. In other words, $H$ is a small regular neighborhood of a Legendrian graph in a contact $3$-manifold (wlog $\mathbb R^3$). Equivalently, $H=(\Sigma\times[0,1],dt+\...
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references on categorification of knot invariants

I am extremely sorry if this is not the right place for this kind of question. I have studied some knot theory, quantum invariants and would like to study more about categorification of knot ...
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202 views

What are the “correct” references for the Vassiliev invariant?

Is there a good survey paper which describes the general ideas of Vassiliev's invariant? I am not an expert on knot theory, many references are too technical for me. Could Vassiliev's invariants be ...
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Is every tricolourable knot chiral?

In other words, can anyone give an example of a tricolourable knot that is equivalent under the Reidemeister moves to its mirror image? If not, is it known or conjectured anywhere in the literature ...
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386 views

$0$-surgeries on trefoil and figure-eight

Let $M$ and $N$ be $3$-manifolds obtained by zero-surgery on (left-handed) trefoil and figure-eight knot respectively. What is the easy way to prove that $M$ and $N$ are not homeomorphic? Note: When ...
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JSJ-type decompositions for knots

According to Wikipedia, JSJ decomposition for 3-manifolds is the following statement: "Irreducible orientable closed (i.e., compact and without boundary) 3-manifolds have a unique (up to isotopy) ...
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1answer
281 views

Harmonic functions on knot complements

In Axler's Harmonic Function Theory, he and his coauthors develop the theory of harmonic functions on spheres and discs by considering the restrictions of arbitrary polynomials on the sphere $S^{n-1} =...
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May this slice disk for the unknot be pushed into the boundary?

Write the 4-ball as $\mathbb{D}^4=\mathbb{D}^2\times \mathbb{D}^2$. Then its boundary $\mathbb{S}^3\simeq \mathbb{S}^1\times \mathbb{D}^2\cup \mathbb{D}^2\times \mathbb{S}^1$. We will use implicitely ...
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2answers
278 views

Rational slice knot that is not slice

Does there exists a knot $K\subset \mathbb{S}^3$ such that $K$ is not slice $\exists W^4$, $\partial W = \mathbb{S}^3$ rational homology ball $\exists $ properly embedded smooth disk $(D,\partial D)\...
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1answer
225 views

unlinking when relaxing the homeomorphism condition

Say that we have two knots $K_1$ and $K_2$ in $S^3$ linked together in $S^3$ and forming the Hopf link. Usually, we can prove that we cannot unlink them by using a link invariant that shows that the &...
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193 views

Invariant knot for finite group actions on $S^3$

Inspired by the Smith conjecture, is there a finite group action on $S^3$ (by smooth or analytic diffeomorphisms) which possesses an invariant knotted circle?

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