# Questions tagged [seifert-surfaces]

Surfaces bounding a knot or a link.

19 questions
Filter by
Sorted by
Tagged with
157 views

### 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 $f: S^1\times[0,1]\to \mathbb R^3$ ...
226 views

### 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 ...
100 views

### 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 ...
196 views

### 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 ...
347 views

509 views

### 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 ...
377 views

### 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 ...
92 views

### 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 ...
1k views

### 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 ...
490 views

### 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 ...
276 views

### 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 ...
598 views

### 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 ...
617 views

### 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 ...
457 views

### 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? ...
589 views

### 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 ...
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 ...