# 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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### Generalized Brunnian links

A Brunnian link of order $n$ is nontrivial link of $n$ rings
that becomes a trivial link of $n-1$ rings if any ring is
removed. They were classified up to link-homotopy by
Milnor in 1954. This ...

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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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### Computing $\pi_2$ of the complement of a 2-knot or spaces with aspherical splittings?

As far as I know, it is an open question if the complement of a ribbon disk $D^2 \subset B^4$ is aspherical. In reading "Some remarks on a problem of J.H.C Whitehead" by Howie, it is noted that there ...

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### Mean curvature flow and knot theory

I am wondering if the mean curvature flow of one-dimensional submanifolds of $\mathbb{R}^3$ is understood well enough to give some perspective on (and hopefully a proof of) something like the Fary-...

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### Is the following link impossible to construct?

I'm wondering if it is possible to prove that the following link as described is impossible to construct, or, if not, to construct it. Consider 6 rings, labeled A through F. I want the system to ...

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### Unknotting algorithm in higher dimensions?

Suppose we are given a 2-knot (say by a movie). Is there an algorithm to tell if it is unknotted ? I suppose that it could matter if I say "topologically" or "smoothly" here since those could be ...

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### Homology torsion in the double branched cover of a tangle?

Let $T$ be a locally unknotted $2$-tangle in $B^3$ and $\Sigma(T)$ be its double branched cover.
Can $H_1(\Sigma(T))$ have a non-trivial torsion? (Obviously, not for rational tangles.)

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### Implementation of Koebe–Andreev–Thurston circle packing?

The circle packing theorem (Koebe–Andreev–Thurston theorem) claims for a planar graph, we can pack disjoint circles, such that: the circles correspond to vertices and the disks are tangent if the ...

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### Kontsevich integral on tangles and Fubini

I am reading about the Kontsevich integral, following this text : https://pdfs.semanticscholar.org/635b/c6370e8aba381724eaaa36abefba7f7a5bec.pdf
At some point (page 10 to be exact) the author claims ...

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### Upper bounds on the genus of the surface produced by Seifert's algorithm

Let $K$ be a knot with genus $g$. Seifert's algorithm produces a surface of genus $k$ whose boundary is $K$. In general $k$ may be larger than $g$, but are there any bounds on how much larger it can ...

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### Integer surgery on $S^3$

I know that any compact orientable 3-manifold can be obtained from the three sphere $S^3$ by an integer surgery. I am not sure why the surgery operation is completely determined by Where we map ...

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### Rigid Brunnian links for $n \geq 4$

Brunnian links consist of $n$ linked un-knot components such that the cutting of any component leaves all components unconnected. The most famous example is the three-component Borromean rings (or ...

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### Connections between spectral geometry and critical point/Morse theory

I am researching electrostatic knot theory, which is essentially the theory of harmonic functions on knot complements. I want to understand the number of critical points of the electric potential, ...

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### Equivalence of conditions for torsions of links to be defined

The torsion of a link complement $S^3 \setminus L$ is defined in terms of the twisted chain complex $C_*(S^3 \setminus L; \rho)$, where $\rho : S^3 \setminus L \to \operatorname{GL}_k(\mathbb{k})$ is ...

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### Does tangle closure determine the triviality of the tangle?

Let $NS(T), D_+(T), D_-(T)$ denote closures of a $2$-tangle $T$ as in the picture. $T_0$ (below) is called trivial.
Question: Suppose that $NS(T)$ and $D_+(T)$ are trivial knots.
Does it imply that $...

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### $\mathbb{Z}/2\mathbb{Z}$ coefficients in gysin sequence

I am reading the article "Signature of links" by Kauffman and Taylor. Here they show that it is possible to calculate the nullity of a link $L\subset S^3$ by knowing the first betti number of the ...

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### Understanding fundamental group of Poincare homology sphere

I'm currently reading Knots, Links, Braids, and 3-Manifolds by V. V. Prasolov and A. B. Sossinsky. I have trouble understanding the following picture. The dashed line denotes a trefoil whose tubular ...

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### Transverse knots with knot types of strongly quasi-positive knots

In 2008, Etnyre and Van Horn-Morris proved that if $L$ is a fibered strongly quasi-positive link, there is a unique (up to transverse isotopy) transverse link with the knot type of $L$ in the standard ...

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### Approximative extension of the autohomeomorphism of the complement of the trivial knot?

Let $S^1\subset \mathbb{R}^3$ be the unit circle and suppose $h\colon \mathbb{R}^3\setminus S^1\to \mathbb{R}^3\setminus S^1$ is a homeomorphism. Clearly it might be that $h$ cannot be extended to $S^...

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### Do there exist any variational principles on the space of braids (or knots)?

This is very speculative question and I do not know where to start looking up the literature, or if what I am looking for is even mathematically possible/meaningful.
Q: I am interested in finding out ...

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### Alexander duality and homology equivalence

While reading the paper of Kauffman and Taylor "Signature of links" I found the following situation.
In the proof of Theorem 2.6 they suppose that two links $L_1, L_2\in \mathbb{S}^3$ are ...

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### An upper bound for second type of Reidemeister move

Suppose there are tow diagram $D_1$ and $D_2$ of knot $K$ with $c_1$ and $c_2$ crossing. Are there any bound of second type of Reidemeister move in term of $c_1$ and $c_2$? In other words, Are there ...

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### Braided monoidal categories

I know that it has been shown that $E_2$ algebra objects in Categories are simply braided monoidal categories.
In particular, Lurie says that an $E_2$-monoidal structure on the infinity-category $N(C)$...

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### “Taylor series” of a knot crossing - does that make sense?

Assume you have a knot "colored" with the irrep of a quantum Lie algebra with its parameter at $q=1+\epsilon$ where $\epsilon$ is small. When it is zero, all crossings turn virtual. Does it make any ...

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### Rational surgery and attaching $2$-handles

It is well-known fact that integral Dehn surgeries on $3$-sphere $S^3$ are viewed as the result on the boundary of attaching $2$-handles $B^2 \times B^2$ to the $4$-ball $B^4$.
Is there an analogue ...

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### When do two knots have isomorphic fundamental bikeis?

A kei, also known as an involutive (or involutory) quandle, is a quandle $(Q,*)$ satisfying the involution condition that $(x*y)*y=x$ for all $x$ and $y$. Just like we can define a fundamental ...

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### Which knots in the Rolfsen knot table are (quasi) positive braid knots?

A knot is called a positive braid knot if it can be presented as the closure of a positive braid. A knot is called a quasi-positive braid knot if it can be presented as the closure of a braid which is ...

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### Reconciling Sullivan's theorem with the hyperbolic structure of the Figure–8 knot complement

I am interested in the 3-manifolds with hyperbolic structures from the physics (gravity) perspective. I encounter this paper https://arxiv.org/pdf/hep-th/9812206.pdf whose Eq. (9) mentions a theorem ...

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### Computation of \tau invariant

I am trying to understand the following inequality, $$0 \leq \tau (K_{+}) - \tau(K_{-}) \leq 1$$ from the following paper by Livingston. \ https://arxiv.org/pdf/math/0311036.pdf . At page 737 , he ...

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### Minimally non-alternating links

I call a non-alternating link "minimally-..." if it can be obtained by taking an alternating link and flipping a crossing. I guess at least all algebraic link have the property. Do you have a proof, a ...

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### Boundary of slice disk exterior is the zero surgery of slice knot

I couldn't exactly guess the level of question. I asked in Math Stack Exchange. (Depending on the situation, I will remove it from here.)
I'm trying to understand a sketch of proof of Livingston and ...

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### How to compute fundamental groups of slice disk complements?

To compute the fundamental group of the complement $S^3 \setminus K$ of a knot, one usually uses the Wirtinger algorithm. Is there a similarly well-established procedure for computing the fundamental ...

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### Link invariants distinguishing components

I was recently thinking about links where each component plays the same role: for every permutation of components, there is an isotopy permuting these components in the prescribed way. In the vein of ...

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### Parametric Seifert surfaces for parametric families of knots in $\mathbb{R}^3$

Let $K_t$ be certain $1-$ parametric family of knots in $\mathbb{R}^3$. I am wandering what are the precise obstructions for a parametric Seifert surface to exist; i.e. a $1-$parametric family of ...

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### Whitney trick on knots

Suppose Reidemeister 2 and 3 moves are valid on a knot/link diagram, but not 1. With the Whitney trick, you can annihilated a positive and a negative writhe. Picture But as you see, one writhe points ...

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### Slice knot and rational ball

I'm trying to understand the proof of the following classical theorem of Casson and Gordon.
Theorem (CG86): If $K$ is a smoothly slice knot in $S^3$, then its double branched cover $\Sigma(K)$ bounds ...

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### Undirected Alexander polynomial (sort of)

Take the skein relation of the Alexander polynomial: $S^1-S^{-1}-zS^0=0$, where z is the parameter of the Alexander polynomial and $S$ the overcross braid element. "Multiply" the equation with ...

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### Khovanov Homology in Macaulay2

Has anyone ever written code for computing Khovanov homology in Macaulay2 or other similar software? I know there are various excellent programs for computing Khovanov homology, but I'm currently ...

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### What is the state of research on finding all Prime Knots with 17 Crossings?

In this 1998 journal paper, all the prime knots with 16 or fewer crossings are found (some of which were found earlier by others). There are over 1.7 million such knots. But the prime knots with 17 ...

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### Which knot invariants have no known diagram-independent descriptions?

Many knot invariants in knot theory are discovered by finding a property of knot diagrams which is invariant under the three Reidemeister moves. Now in principle, any knot invariant can be described ...

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### Can different knots have the same numbers of quandle colorings for all quandles?

Let $K_1$ and $K_2$ be two knots such that for all finite quandles $X$, the number of colorings of $K_1$ by $X$ is the same as the number of colorings of $K_2$ by $X$. Then my question is, must $K_1$ ...

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### Easy lemma for trivalent graphs in colored Jones polynomial

In his 2008 paper,
Tanaka, Toshifumi, The colored Jones polynomials of doubles of knots, J. Knot Theory Ramifications 17, No. 8, 925-937 (2008). ZBL1149.57023.
Tanaka stated a lemma (Lemma 3.3) ...

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### Knot and its embedded disk

Let $K \subset S^3$ be an arbitrary knot. Let $D$ denote the embedded disk in $B^4$ bounded by $K$.
Up to diffeomorphism, is it possible to describe the followings (at least for some trivial knots, ...

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### Tracking down an elusive book

A few weeks ago I had a very engaging talk with a faculty member, where he told me lots of interesting things about quantum algebras, know theory and Reshetikhin-Turaev invariants (this field is not ...

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### Distinguishing Square Knot and Granny Knot using Quandles

It is known that the square knot and the granny knot are nonequivalent although they have isomorphic fundamental groups.
I want to write a work on knot theory and provide these knots as an example ...

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### Reference request: A knot is tame if and only if it has a tubular neighbourhood

Definitions:
A knot is an embedding $\kappa:S^1\hookrightarrow S^3$ (we do not require smooth or polygonal).
Two knots $\kappa,\,\lambda:S^1\hookrightarrow S^3$ are equivalent if one of the following ...

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### Why does the longitude correspond to Frobenius in Arithmetic Topology, and other strange phenomena

I am trying to adress Morishita's book Knots and Primes to discover a bit about Arithmetic Topology, but some analogies puzzle me. I know that the correspondence should be addressed with a grain of ...

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### Finding a presentation matrix with low dimension

Let $R=\mathbb Z[t^{\pm}]$ and $M$ a finitely generated $R$-module. With $A$ a presentation matrix, i.e we have the following exact sequence (usually I'm working with the case where $A$ is an square ...

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### Are Turaev-Viro invariants holonomic?

Consider a 3-manifold $M$ with a boundary, which is a genus $g\geq 1$ surface $\Sigma$. Fix a triangulation $T$ of $\Sigma$. Then Turaev-Viro invariants $TV_q(M)$ are functions, assigning to integer ...

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### What are the possible linking matrices of a quasi-positive link?

I was surprised recently to come across a 3-component link where the linking number of two of the components was negative. For a while I thought I had made a mistake, then I thought a little more and ...