Questions tagged [seifert-surfaces]
Surfaces bounding a knot or a link.
22 questions
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compressibility of Seifert surface after 0-surgery
Gabai's solution of the Property R conjecture shows that a minimal genus Seifert surface of a knot, capped off in the 0-framed surgery along that knot, is of minimal genus in its homology class. In ...
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Is there an algorithm for the genus of a knot?
A Seifert surface of a knot is a surface whose boundary is the knot. The genus of a knot is the minimal genus among all the Seifert surfaces of the knot. My question is, is any algorithm known to ...
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Minimal area of Seifert surfaces
Let $K$ be a knot smooth knot in a 3-manifold $M$ and fix a metric on $M$. Let $F$ be a orientable surface of genus $g$ with one boundary component. Then we can consider the family of all maps $\...
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Is every virtual knot group an HNN extension?
A basic fact in knot theory is that a knot group $\pi(K)$ is an HNN extension of $\pi(F)$, the fundamental group of a Seifert surface complement. A nice discussion of this may be found in Chapter 11 ...
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Seifert surfaces of torus knots
Does anyone know a nice description of a Seifert surface of a torus knot? I can construct such surfaces in band projection, but what I get is ugly and unwieldy. Is there some elegant description for ...
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Seifert surfaces via Alexander duality
If we take a knot $K$ in $S^3$, there are several ways to construct the associated Seifert surface. One way, which I am not familiar with, I just came across in a paper I am reading. It goes like ...
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Original proof of the existence of Seifert surfaces
I read on Wikipedia that Frankl and Prontrjagin were the first to prove that a link $\mathbb{R}^3$ bounds a surface. A few years later Seifert published a proof using the "Seifert algorithm" which ...
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Is there a notion of "ribbon 2-category"?
It there some notion of ribbon 2-category, which would allow for, say, talking about the Seifert surface of links (which is a 1-morphism in some ribbon category) as a 2-morphism in the category?
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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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Seifert genus of the lift of a knot in its cyclic branched covers
I was wondering if there are any known examples of knots $K$ in $S^3$ with Seifert genus $g$ so that the lift of $K$ sitting inside its $n$-fold cyclic branched cover bounds an embedded surface of ...
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Essential surfaces in knot complements
Given any knot $K \subset \mathbb{S}^3$, one can find a closed oriented embedded surface $S$ such that $K \subset S \subset \mathbb{S}^3$. Moreover, pick such an $S$ that has minimal genus.
One can ...
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Are two slice surfaces with minimal genus isotopic?
For a link $L\subset S^3$ and two Seifert surfaces (edit: a better name would be slice surfaces as the comments below 1 2 point out) with minimal genus $S_1,S_2\subset B^4$, I have the following ...
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Are there examples of different knots with identical Jones polynomials and different Seifert Genus?
I had asked this question on math.stackexchange 2 days back but came up empty handed so I wanted to ask it here.
Are there known examples of $2$ non equivalent knots that have identical jones ...
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Pre-images of Seifert surfaces are incompressible?
Consider a knot $K \subset S^3$ and let $M_K$ be the associated double branched cover. The pre-image $S$ of a Seifert surface is a surface without boundary inside $M_K$.
Can $S$ be incompressible? If ...
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How disconnected can a Seifert surface be?
Seifert surfaces
The standard definition of a Seifert surface for a link in $S^3$ is an oriented, compact surface embedded in $S^3$, bounding the link. Often, it is assumed to be connected, but given ...
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Immersed Seifert surfaces of minimal genus
Let $K\subset S^3$ be a knot. We denote by $X=S^3\setminus \nu K$ the knot exterior, i.e. the complement of an open tubular neighborhood of $K$. An immersed Seifert surface for a knot $K$
is an ...
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Linking circles inside an immersed surface
(Migrated from Math Stack Exchange)
A smooth embedding $f : D \to \mathbb{R}^3$ can be isotoped to a canonical inclusion $D \hookrightarrow \mathbb{R}^3$. (This is part of a proof that only the ...
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Minimal genus of Seifert surface of torus knot
Let $(p,q)$ be a pair of coprime (positive) integers. Consider the torus knot $T_{p,q}$. What is the minimal genus of an (embedded) oriented Seifert surface for this knot?
It is not had to convince ...
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An equivalence relation on knots similar to concordance
Let $L_1$ and $L_2$ be two nonintersecting picewise-linear or smooth knots in $\mathbb R^3$. Suppose they are ambient isotopic. Does there exist an embedded surface $f: S^1\times[0,1]\to \mathbb R^3$ ...
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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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Trefoil Knot Seifert Minimal Surface Equation
I am not very familiar with knot theory nor with minimal surfaces, so I already apologize if my question appears too naive or simple :). I am trying to do the following:
Starting from a real ...
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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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