Questions tagged [knot-theory]
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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hyperbolic 3-manifold of finite volume
Is there a complete description of hyperbolic 3-manifold of finite volume ?
Or similarly a classification of finitely generated torsion free subgroups of $PSL(2,\mathbf{C})$ with finite covolume?
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On the quadratic equivalence of fields
I have spent the past two years studying abstract Witt rings. These objects are a generalization of "The Witt ring of a field," an algebraic invariant of fields of characteristic not equal to 2. ...
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On the genus of thin knots and the degree of the Alexander polynomial
I came across this papper by JA Baldwin which presents a combinatorial definition for the knot Floer homology.
At a certain paragraph of the third page the author makes the next statement: the genus ...
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Gap in Przytycki's computation of the skein module of links in a handlebody?
I am reading the paper [1], where the author proves that the skein module of links in a handlebody $F\times I$ has a free basis given by products $D_1 \cdots D_n$ where each $D_i$ is the closure of $...
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Braid group and knot group
It is well known that braid groups and knot groups share many common properties. For example, they have the same $H_1$ and they are both residually finite and (hence) Hopfian. On the other hand, we ...
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Are there two non-equivalent exotic structures on $\mathbb{R}^4$ coming from topologically slice, non-slice knots?
For a knot $K \subset S^3$, which is topologically slice but not slice (in a smooth way), there's a four manifold $\mathbb{R}^4_K$, homeomorphic but not diffeomorphic to standard euclidean $\mathbb{R}^...
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Relatively hyperbolic knot groups
I am looking for references on the geometry of knot groups. For instance, I am interested in the following question:
When is a knot group relatively hyperbolic?
Hyperbolic knots are known to have ...
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Knotted TSP tours in 3D?
In the plane, the Euclidean TSP tour never crosses itself—it is always a simple polygon.
I am wondering if there is a similar constraint for the Euclidean TSP tour
of points in $\mathbb{R}^3$.
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Non Seifert incompressible surfaces detected by ideal points
Given a 3-manifold with toric boundary, the Culler-Shalen theory associates an incompressible surface to any ideal point of its character variety. From the proof of the Neuwirth conjecture, one knows ...
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How does the topology of the graphs' Riemann surface relate to its knot representation?
Let's consider the following bipartite cubic planar non-simple graph
$\hskip2.3in$
Looking at the orientation of the edges around the vertices, it is obvious that left and right are oriented opposite.
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Why is Khovanov homology considered a 'categorification'?
I understand that the Euler characteristic of Khovanov homology is the Jones polynomial. But in what sense does this give category theory structure to the Jones polynomial, i.e., what are the objects ...
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A question about so-called "Brunnian Links" or "Brunnian Rings"
Let E(3) be 3-dimensional Euclidean space with its standard topology. Brunnian Rings are subsets of E(3) which are simple closed curves. It is known that an arbitrarily large set S(3) of these ...
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Regular projection of a link, proof in the smooth category
Given two $C^1$ immersed curves $f, g: S^1 \to {\mathbb R}^3$ with disjoint image, I would like a simple proof, working only in the smooth category, that there exists a unit direction $y \in {\mathbb ...
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Classification of pretzel links up to link homotopy using alexander quandle
I am currently reading this paper where the author classifies the pretzel links up to link homotopy using a quasi-trivial quandle $\mathbb{Z}_{k}[t^{\pm 1}]\diagup_{(t-1)^{2}}$, and I find it ...
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Invariance of Khovanov homology under first Reidemester move
I am studying Khovanov homology from five lectures on Khovanov homology
and I want to try to show Khovanov homology is invariant under first Reidemester move but I cannot understand how we can write
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Linking number a complete invariant of link homotopy
I read in Milnor's article "Link groups", where he defines invariants to classify links up to link homotopy, that the linking number is a complete invariant which can tell almost trivial two ...
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Higher-dimensional version of Fenchel's theorem that total curvature is $\ge 2 \pi$
Q. Is there a higher-dimensional version of the theorem due to Fenchel that the
total curvature of a closed curve in $\mathbb{R}^3$ is $\ge 2\pi$,
with equality only if the curve is planar and convex?
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1-Bridge Braids in Solid Tori (Berge-Gabai knots), dual knots, & knot exteriors
Let $K$ be a knot in a solid torus. Combining results of Berge and Gabai, we know that if $K$ admits a solid torus surgery, then $K$ is a 1-bridge braid. Using Gabai's result, we can figure out what ...
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What is the preimage of a braid in a covering space branched over the braid?
For a knot $K\subset \mathbb{S}^3$, one can construct the covering space branched over that knot by assigning elements of the symmetric group $S_n$ to each arc of the knot. You can find the knot group ...
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The singularity type of a non-torus link
It is a well-know result that singular surfaces like
$$x^p+y^q=0$$
(for complex $x$ and $y$) can be associated with $(p,q)$-torus links by considering the intersection of this surface with a small ...
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How to braid a ribbon knot
Is there any algorithm known for braiding ribbon knots? More specifically I need to braid a generic ribbon knot presented as boundary of a ribbon surface= union of two 0-handles and one 1-handle. (...
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Monopole Floer Homology vs. Heegaard-Floer theory
I have a (possibly very naive) question: what is the relation between Monopole Floer Homology and Heegaard-Floer theory? (both known and conjectured)
Is there some version of Atiyah-Floer conjecture ...
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Can Khovanov homology have arbitrarily large torsion?
Can Khovanov homology have arbitrarily large torsion?
That is, given $N\gg 0,$ does there exist $k>N$, a knot (diagram) $D$ and $i,j \in \mathbb{Z}$ such that $\operatorname{Kh}^{i,j}(D) = \mathbb{...
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Seiberg-Witten theory in 4d is categorification of Seiberg-Witten in 3d
According to Gukov et al. in this 2017 paper Seiberg-Witten theory in 4d categorifies Seiberg-Witten theory in 3d. In what sense is this phrase mentioned? I know what the process of categorification ...
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Upsilon of an alternating knot
I have a couple of questions about how Oz-Stip-Sz computes the upsilon function invariant of an alternating knot in their upsilon ($\Upsilon$) paper here.1 This is theorem 1.14 (on the bottom of page ...
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Calabi-Yau manifolds and knot theory
In the paper "The Volume Conjecture and Topological Strings" it is said that the mirror Calabi-Yau threefold is given by
$X := \{ (x,y,u,v) \in \mathbb{C^* \times\mathbb{C^*} \times \mathbb{C} \...
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Plat representations of unlinks
Suppose that $\beta$ is a $2n$-strand braid with plat closure $L$. We can multiply $\beta$ on either side by a member of the Hilden subgroup to get a new braid whose plat closure is still $L$. Or we ...
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braid representation of a Montesinos link
Is it possible to get a braid representation for a general Montesinos link with small number of strands? I know by Alexander's theorem it is possible to braid any link but is it possible to find a ...
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Criteria for existence of basis for Seifert surface that has trivial linking with other component of link
Say we have a 2-component link $L$ with components $L_1$ and $L_2$. Are there known conditions that will ensure that there exists a Seifert surface $S$ of $L_1$ with curves $\alpha_1,\beta_1,...,\...
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Lower bounds on the bridge number of a knot (and tabulation of bridge numbers)
What methods do we have for finding lower bounds on the bridge number of a knot?
(I'm aware of this thread, but the question does not seem exhausted there. For example, how can the Jones polynomial ...
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On the geometrization of double branched covers
I recently got into Lickorish's paper Prime knots and tangles and a question, which I didn't have the first time I read it, naturally emerged.
The Thurston-Perelman Geometrization Theorem asserts ...
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Is there a polynomial-time algorithm for untangling the unknot?
I've found assertions that recognising the unknot is NP (but not explicitly NP hard or NP complete). I've found hints that people are looking for untangling algorithms that run in polynomial time (...
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List of Gauss Codes?
I'm looking for a list of Gauss codes for knots 12 crossings and higher. I downloaded the list of all the knots up to 12 crossings from Charles Livingston's Knot Info. Are there any other online ...
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Certain flow on space of lattices in $\mathbb C$
It's pretty obvious that the space $L$ of unimodular lattices $\Lambda \subset \mathbb C$ is a complement of trefoil knot: $S^3 \setminus T$. Consider a flow $f_t:= \left( \begin{array}{ccc}
e^t & ...
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Elementary proof that knot complements are path-connected
The complement of any (topological) knot is path-connected. More precisely, if $K$ is a subset of $\mathbb{R}^3$ (or $S^3$) homeomorphic to $S^1$, then $\mathbb{R}^3\setminus K$ (or $S^3\setminus K$) ...
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Knot complements with respect to Thurston's 8 geometries
We can put knot complements in three buckets: hyperbolic knots, satellite knots, and torus knots.
Can this classification be made with looking at the (complete) metric on the complement, if it has ...
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Structure of Gordian graph of knots
The Gordian graph of knots has the knot isotopy classes as it's vertices, and an edge whenever you can pass from one knot to a other via a "finger move", equivalently if for some diagram of the knot ...
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Complement of figure 8 knot - zero vertex [closed]
Thurston in "3-dimensional Geometry and Topology" explicitely creates the hyperbolic complement of the figure 8 knot by glueing two ideal tetrahedra.
What I do not understand is that he pushes the ...
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Characterizing different chain links
Imagine I have 3 rings (or any other closed line) which cannot twist (by twist I mean as in the Whitehead link, for example). There are 3 ways to link them, one of which is the Borromean rings ...
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Vertex deletion in knotless and linkless graphs
If $G$ is a graph and $G-v$ is linkless for some vertex $v$, is $G$ necessarily knotless?
Of course, one can assume that $v$ is adjacent to every vertex in $G-v$.
Here, a graph is linkless if it ...
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What is the most symmetric configuration of four 2-surfaces linked in $S^4$?
What are some of the most symmetric configurations of four 2-surfaces linked in the 4-dimensional sphere $S^4$?
To make a lower-dimensional analogy, recall that in 3-dimensional sphere $S^3$, we can ...
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Knot invariants from skein relations
Let $k$ be a fixed field. A skein relation defined on $k$ is a local relation on $k$-linear combination of oriented planar link diagrams saying that one can trade an overcrossing for an undercrossing ...
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How many different knot types can have the same shadow (projection)
If we take a knot diagram and ignore the over/under-crossing information, we obtain a shadow (or projection). Thus a shadow is simply a plane embedding of a 4-regular graph. Clearly, two non-...
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A link of four 2-tori $T^2$ in $S^3 \times S^1 \# S^2 \times S^2 \# S^2 \times S^2$
Step 1: We glue two sets of complement space of $D^2\times T^2$ out of the 4-sphere $S^4$, through their $T^3$ boundary to obtain a new 4-manifold:
$$(S^4 \smallsetminus D^2\times T^2) \cup (S^4 \...
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Coloring Snarks
I was looking at coloring a representation of a snark graph as a drawing with crossings. I colored the arcs with two rules: if two arcs meet at the same vertex they have a distinct color. If three ...
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What is this type of knot algebra called within knot theory?
https://youtu.be/co78AEqsv3s?t=1901
Within this video at 31:41, Professor Ronald Brown handles knots algebraically, I am confused to how he deals with the crossing indices. He substitutes and ...
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$S^3$ as cyclic branched cover of itself
In Chapter One of his notes (March 2002) Thurston says:
If $K$ is the trivial knot the cyclic branched covers are $S^3$. It seems intuitively obvious (but it is not known) that this is the only ...
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Unknotting knot diagrams by Reidemeister moves and crossing changes
It is well known that any knot diagram can be unknotted by a sequence of crossing changes (i.e., changing an overcrossing with an undercrossing or vice versa) and of Reidemeister moves. More precisely,...
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Structure of foliations of codimension 2 on three dimensional torus
Is it possible to have a one-dimensional foliation on three dimensional torus such that the foliation has a trefoil knot as its leaf?
Moreover, does a one dimensional foliation on three dimensional ...
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Applications of arithmetic topology to number theory
There is a well-known analogy between 3-manifolds and number fields, with knots corresponding to prime ideals. Are there any results in number theory that have been proven using topology through this ...