Knots in Manifolds: Beyond Euclidean Space — Epoche B2
Knots in Manifolds: Beyond Euclidean Space Classical knot theory asks a question that already presupposes an answer: given a loop in $\mathbb{R}^3$, is it tangled? All the difficulty is taken to live in the loop, and none in the space, because $\mathbb{R}^3$ is assumed to be a passive stage on which the curve performs. That assumption is doing more work than it looks. Change the ambient space and a perfectly round circle — a curve with no crossings at all — can become impossible to shrink to a point, while the machinery that classical knot theory relies on, beginning with the Seifert surface [1] , can simply cease to exist. This essay works out what a knot's ambient 3-manifold contributes: which invariants survive intact, which survive only in weakened form, and which of the familiar constructions turn out to have been consequences of a property of $\mathbb{R}^3$ that nobody bothered to state. What are knots and manifolds? Topology studies the properties of geometric objects preserved under continuous deformations — stretching, bending or twisting, but not tearing or gluing. From a topological perspective a coffee cup and a doughnut are equivalent, since one can be continuously deformed into the other. A manifold is a space that locally resembles Euclidean space. The surface of the Earth is a 2-manifold, because any small patch of it looks like a piece of $\mathbb{R}^2$. Here we work with 3-manifolds , spaces that locally look like $\mathbb{R}^3$; examples are $\mathbb{R}^3$ itself, the 3-sphere $S^3$, and the solid torus. A knot $K$ is a smooth embedding of the circle $S^1$ into a 3-manifold $M$. An embedding is a map that is injective (no self-intersections) and continuous, with continuous inverse on its image. A knot is trivial , or unknotted , in $M$ if it can be moved by a continuous deformation within $M$ onto a circle bounding an embedded disc in $M$. Nothing in this definition mentions crossings or diagrams; those belong to $\mathbb{R}^3$, and the first thing to notice is how little of the theory survives without them. It is also worth recording that $\mathbb{R}^3$ and $S^3$ give the same theory: adding one point at infinity to $\mathbb{R}^3$ produces $S^3$, and a knot is trivial in one exactly when it is trivial in the other, so classical knot theory may be read as taking place in either. Contractibility and simply connected spaces A loop in a manifold $M$ is a continuous map from $S^1$ into $M$ — note that a loop need not be injective, so every knot is a loop but not conversely. A loop is contractible if it can be continuously shrunk to a single point within $M$. Imagine a rubber band lying on a surface: if you can shrink it to a dot without lifting it off the surface, it is contractible. A manifold $M$ is simply connected if every loop in it is contractible. Euclidean space $\mathbb{R}^3$ is simply connected, and so is $S^3$. This is the unstated hypothesis behind the classical picture: because every loop can be shrunk, no curve in $\mathbb{R}^3$ is knotted for a reason having anything to do with the surrounding space, and all knottedness must come from how the curve interweaves with itself. An annulus, a flat ring bounded by two concentric circles. A blue loop lies entirely within the ring and can be continuously shrunk to a point without leaving it. An amber loop runs all the way round the central hole; any attempt to shrink it would have to cross the hole, which is not part of the space, so it is non-contractible. contractible non-contractible annulus: the 2-dimensional analogue Fig. 1 — An annulus, the simplest space with a hole. The blue loop lies inside the ring and shrinks to a point without ever leaving it. The amber loop encircles the hole; shrinking it would require passing through territory that is not part of the space, so it is non-contractible. Both are round circles, geometrically identical apart from position and size — the difference between them is contributed entirely by the space, not by the curve. The fundamental group To record the presence of holes we use the fundamental group . Two loops based at a common point $x_0$ in $M$ are homotopic if one can be continuously deformed into the other while keeping the base point fixed. The set of homotopy classes of such loops, with concatenation (follow one loop, then the other) as the group operation, forms the fundamental group $\pi_1(M, x_0)$. For a path-connected space the base point does not affect the isomorphism class, so we write $\pi_1(M)$. A trivial fundamental group means the space is simply connected. Three examples: Euclidean space is simply connected, so its fundamental group contains only the identity $e$: $$ \pi_1(\mathbb{R}^3) = \{e\}. $$ For the circle $S^1$ the fundamental group is the integers, each integer being the number of times a loop wraps round and in which direction: $$ \pi_1(S^1) \cong \mathbb{Z}. $$ A solid torus $D^2 \times S^1$, a filled-in doughnut with $D^2$ the disc cross-section and $S^1$ the circle it is swept along, deformation-retracts onto its core circle and so has the same fundamental group: $$ \pi_1(D^2 \times S^1) \cong \mathbb{Z}. $$ A knot has no base point, so what it determines is a free homotopy class — equivalently a conjugacy class in $\pi_1(M)$ rather than a single element. That refinement costs nothing here, because the criterion only needs the class to be non-trivial, and conjugacy preserves triviality. If $K \subset M$ is not freely null-homotopic, it cannot be shrunk to a point in $M$, and it is therefore non-trivial in $M$ — whatever it may look like. The core circle of a solid torus is the standard illustration: as a curve in $\mathbb{R}^3$ it is a round circle and the plainest possible unknot, but inside the solid torus it generates $\pi_1 \cong \mathbb{Z}$ and cannot be contracted at all. Its knottedness is entirely the space's contribution. Homology, and when a Seifert surface exists The fundamental group is powerful but often non-abelian and hard to comp