Differential characterization of unknots

Question

How can the closed simple curves in $\mathbb{R}^3$ be characterized that can be boundaries of a 2-dimensional oriented surface in $\mathbb{R}^3$? Intuitively I would tend to say that it's exactly the family of unknots. But even if so: (how) can this family of curves be characterized - maybe by means of differential geometry, by curvatures and torsions?

Answer 1

Every simple closed curve in $\mathbb{R}^3$ (not only the unknots) is the boundary of some compact, connected, oriented surface. Such a surface is called a Seifert surface of the curve; Seifert wasn't the first to prove this fact, but he did provide an elementary algorithm to construct Seifert surfaces. You can find the details of Seifert's Algorithm in any knot theory text.

Answer 2

Every curve (knotted or not) is the boundary of a regular surface. See here for an image of a surface bounding the simple knot.

What should be true, is that unknotted curves are exactly the curves which are boundaries of embedded disks (the disks should not cross the curve).

Answer 3

Since the "boundary of oriented surface" version was answered, I address the question of geometric characteristics of unknots.

There are geometrically natural sufficient conditions for a closed curve to be an unknot. One of them is in terms of the following measure of distortion, introduced by Gromov in the 1980s.

Definition. The distortion of an embedding $\gamma: S^1\to\mathbb R^3$ is $$\delta(\gamma)=\sup_{s,t\in S^1}\frac{d_\gamma(\gamma(s),\gamma(t))}{|\gamma(s)-\gamma(t)|}$$ Here $d_\gamma$ is the intrinsic distance on the image of $\gamma$, i.e., the length of shortest subarc with endpoints $\gamma(s),\gamma(t)$. And $|\gamma(s)-\gamma(t)|$ is the Euclidean distance (length of a chord).

Gromov showed that $\delta(\gamma)\ge \pi/2$ for any $\gamma$, and asked whether every knot is isotopic to $\gamma$ with $\delta(\gamma)\le 100$ (or some other universal constant).

Motivated by this question, Elizabeth Denne and John M. Sullivan proved that

(i) $\delta(\gamma)\ge 5\pi/3$ for every nontrivial tame knot

(ii) the trefoil knot can be realized with $\delta<7.16$

As far as I know, the gap between (i) and (ii) has not been bridged. And ideally, the result should hold without the tameness assumption.

Recently, John Pardon answered Gromov's question by showing that there are knots that require arbitrarily large distortion:

$\delta(\gamma)\ge \frac{1}{160}\min(p,q)$ when $\gamma$ is isotopic to the $T_{p,q}$ torus knot.

Both papers can also be found on arXiv: Denne-Sullivan, Pardon.

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Along these lines, I mention Gehring's link problem (solved long ago). If $\gamma_1$ and $\gamma_2$ are two disjoint closed curves in $\mathbb R^3$ and $$\operatorname{dist}(\gamma_1,\gamma_2)\ge \frac{1}{2\pi}\min(\operatorname{length}(\gamma_1),\operatorname{length}(\gamma_2))\tag{1}$$ then $\gamma_1$ and $\gamma_2$ are not linked. (They can still be knotted individually). The bound in (1) is sharp, since the Hopf link can be realized by two circles of unit radius at distance $1$ from each other.