When k looks and smells like the unknot…

Conan5 Comments

Valentine’s day special issue~ ^_^

Professor Gabai decided to ‘do some classical topology before getting into the fancy stuff’ in his course on Heegaard structures on 3-manifolds. So we covered the ‘loop theorem’ by Papakyriakopoulos last week. I find it pretty cool~ (So I started applying it to everything regardless of whether a much simpler argument exists >.<)

Let M be a three dimensional manifold with (non-empty) boundary. In what follows everything is assumed to be in the smooth category.

Theorem: (Papakyriakopoulos, ’58)
If f: \mathbb{D}^2 \rightarrow M extends continuously to \partial \mathbb{D} and the image f(\partial \mathbb{D}) \subseteq \partial M is homotopically non-trivial in \partial M. Then in any neighborhood N(f(\mathbb{D})) we can find embedded disc D \subseteq M such that \partial D is still homotopically non-trivial in \partial M.

i.e. this means that if we have a loop on \partial M that is non-trivial in \partial M but trivial in M, then in any neighborhood of it we can find a simple loop that’s still non-trivial in \partial M and bounds an embedded disc in M.

We apply this to the following:

Corollary: If a knot k \subseteq \mathbb{S}^3 has \pi_1(\mathbb{S}^3 \backslash k) = \mathbb{Z} then k is the unknot.

Proof: Take tubular neighborhood N_\varepsilon(k), consider M=\mathbb{S}^3 \backslash \overline{N_\varepsilon(k)}, boundary of M is a torus.

By assumption we have \pi_1(M) = \pi_1(\mathbb{S}^3 \backslash k) = \mathbb{Z}.

Let k' \subseteq \partial M be a loop homotopic to k in N_\varepsilon(k).

Since \pi_1(M) = \mathbb{Z} and any loop in M is homotopic to a loop in \partial M = \mathbb{T}^2. Hence the inclusion map i: \pi_1(\mathbb{T}^2) \rightarrow \pi_1(M) is surjective.

Let l \subseteq \partial M be the little loop winding around k.

It’s easy to see that i(l) generates \pi_1(M). Hence there exists n s.t. k'-n \cdot l = 0 in \pi_1(M). In other words, after n Dehn twists around l, k' is homotopically trivial in M i.e. bounds a disk in M. Denote the resulting curve k''.

Since k'' is simple, there is small neighborhood of k'' s.t. any homotopically non-trivial simple curve in the neighborhood is homotopic to k''. The loop theorem now implies k'' bounds an embedded disc in M.

By taking a union with the embedded collar from k to k'' in N_\varepsilon(k):

We conclude that k bounds an embedded disc in \mathbb{S}^3 \backslash k hence k is the unknot.

Establishes the claim.

Happy Valentine’s Day, Everyone! ^_^

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