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Poincaré conjecture - diagrammatic formulation

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Introduction

The Poincaŕe conjecture can be re-formulated as a conjecture concerning link diagrams. After recalling some preliminaries, we present this diagrammatic formulation.

Kirby equivalence

The framed Reidemeister moves on a link diagram are depicted here.

The Kirby moves on a link diagram are depicted here.

A pair of link diagrams are Kirby equivalent if there is a finite sequence of framed Reidemeister moves and Kirby moves taking one to the other.

Fundamental theorems on 3-manifolds

We shall rely on the following fundamental theorems, which allow for a diagrammatic approach to the Poincaré conjecture.

Theorem (Lickorish-Wallace)

Let MM be a closed, connected, orientable 3-manifold. There is a link diagram LL such that MM is isomorphic to the 3-manifold obtained by the integral Dehn surgery on LL in S 3S^{3} with respect to the blackboard framing of LL.

Theorem (Kirby)

Let M 0M_{0} and M 1M_{1} be closed, connected, orientable 3-manifolds. Let L 0L_{0} (respectively L 1L_{1}) be a link diagram such that the 3-manifold obtained by the integral Dehn surgery on L 0L_{0} (respectively L 1L_{1}) in S 3S^{3} with respect to the blackboard framing of L 0L_{0} (respectively L 1L_{1}). Then M 0M_{0} is isomorphic to M 1M_{1} if and only if L 0L_{0} and L 1L_{1} are Kirby equivalent.

Let LL be a link diagram, with some choice of orientation. We denote the free group on the arcs of LL by F(L)F(L).

We define π 1(L)\pi_{1}(L), the fundamental group of LL, to be the quotient of F(L)F(L) by the normal subgroup generated by words of the form a 3 1a 2 1a 1 1a 2a_3^{-1} a_2^{-1} a_1^{-1} a_2, for any crossing of LL as depicted here, irrespective of the orientation of the horizontal arcs.

Let LL be a link diagram, with some choice of orientation. The longitude of a component of LL is defined to be the word ww which we obtain after carrying out the following procedure.

  1. Pick any arc of LL, say aa. Let ww be the empty word.
  2. Walk around LL, following the orientation. When we walk under an arc bb, whether or not bb belongs to same component of LL or a different one, we add bb to the end of ww if the configuration of orientations at the crossing is as depicted in the first figure here, and add b 1b^{-1} to the end of ww if the configuration of orientations at the crossing is as depicted in the second figure here.
  3. Stop when we return to the arc we started with, namely aa.

The following is a consequence of the van Kampen theorem.

Proposition

Let MM be a closed, connected, orientable 3-manifold. Let LL be a link diagram such that the 3-manifold obtained by the integral Dehn surgery on LL in S 3S^{3} with respect to the blackboard framing of LL is isomorphic to MM. Then the group π 1(M)\pi_{1}(M) is isomorphic to the group π 1(L)/l 1,,l n\pi_{1}(L) / \langle l_1, \ldots, l_n \rangle, where l 1l_1, \ldots, l nl_{n} are the longitudes of the components of LL, and l 1,,l n\langle l_1, \ldots, l_n \rangle is the normal subgroup generated by these.

Diagrammatic formulation of the Poincaré conjecture

Let MM be a closed, connected 33-manifold. The Poincaré conjecture is that if π 1(M)\pi_{1}(M) is trivial (that is to say, isomorphic to a group with one element), then MM is isomorphic to S 3S^{3}.

If π 1(M)\pi_{1}(M) is trivial, then MM is orientable. It thus follows from the Lickorish-Wallace theorem, the Kirby theorem, the preceding proposition, and the fact that integral Dehn surgery on the empty link diagram gives S 3S^{3}, that the Poincaré conjecture is equivalent to the following: if a link diagram LL has the property that the group π 1(L)/l 1,,l n\pi_{1}(L) / \langle l_1, \ldots, l_n \rangle is trivial, then LL is Kirby equivalent to the empty link diagram.

Last revised on June 8, 2018 at 06:02:53. See the history of this page for a list of all contributions to it.