Topological invariants for 3-manifolds using representations of mapping class groups. II: Estimating tunnel number of knots.

*(English)*Zbl 0823.57004
Sally, Paul J. jun. (ed.) et al., Mathematical aspects of conformal and topological field theories and quantum groups. AMS-IMS-SIAM summer research conference, June 13-19, 1992, Mount Holyoke College, South Hadley, MA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 175, 193-217 (1994).

[For Part I see Topology 31, No. 2, 203-230 (1992; Zbl 0762.57011).]

The main result is an inequality giving a lower bound for the tunnel number of a knot in a closed oriented 3-manifold. The key argument uses Witten’s 3-manifold invariants based on a projective unitary representation of the mapping class group of a closed oriented surface occurring in a Heegaard splitting. For a knot in \(S^ 3\) the inequality can be expressed in terms of special values of the Jones polynomial of the knot.

The last paragraph considers the symmetry principle for invariants in the case \(\text{sl} (n,\mathbb{C})\). A formula for the computation of these invariants is derived. For homology spheres obtained by \(1/a\)-Dehn surgery on a knot in \(S^ 3\) the \(\text{sl}(n,\mathbb{C})\)-invariants are shown to satisfy a certain periodicity law.

The relevant tools of conformal field theory, and approaches to Witten’s invariants by Dehn surgery and Heegaard splittings are introduced in the preceding paragraphs.

For the entire collection see [Zbl 0801.00049].

The main result is an inequality giving a lower bound for the tunnel number of a knot in a closed oriented 3-manifold. The key argument uses Witten’s 3-manifold invariants based on a projective unitary representation of the mapping class group of a closed oriented surface occurring in a Heegaard splitting. For a knot in \(S^ 3\) the inequality can be expressed in terms of special values of the Jones polynomial of the knot.

The last paragraph considers the symmetry principle for invariants in the case \(\text{sl} (n,\mathbb{C})\). A formula for the computation of these invariants is derived. For homology spheres obtained by \(1/a\)-Dehn surgery on a knot in \(S^ 3\) the \(\text{sl}(n,\mathbb{C})\)-invariants are shown to satisfy a certain periodicity law.

The relevant tools of conformal field theory, and approaches to Witten’s invariants by Dehn surgery and Heegaard splittings are introduced in the preceding paragraphs.

For the entire collection see [Zbl 0801.00049].

Reviewer: G.Burde (Frankfurt / Main)

##### MSC:

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |

57N10 | Topology of general \(3\)-manifolds (MSC2010) |

20F36 | Braid groups; Artin groups |