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**Unknotting number, genus and companion tori.**
*(English)*
Zbl 0616.57003

We apply the Gabai sutured manifold machine to study the effect of crossing changes on the genus of a knot. Our results are of two types:

First, we show that there are knots for which a crossing change which lowers the unknotting number raises the genus. In fact the argument seems, in principle, to produce a myriad of such examples.

Second, we examine the effect of companion tori on crossing changes. We discover that either such companion tori are essentially unrelated to the part of the knot on which the crossing change is being made, or there is an inequality which relates the genus of the knot before and after the crossing change.

The precise statement and some corollaries are:

Definition: Let K be a knot in \(S^ 3\) and D be a disk which intersects K in precisely two points, of opposite orientation. Let \(L=\partial D\) be given a \(\pm 1\)-framing. Then L is a crossing link and D is its crossing disk. Denote by K’ the knot in \(S^ 3\) obtained from K by doing \(\pm 1\) surgery to L. (K and K’ are related by a single crossing change.) Let T be an incompressible torus in \(S^ 3-K.\)

Theorem: If genus(K’)\(\leq genus(K)-2\) then any companion torus of K can be isotoped disjoint from L.

Corollary: If K is a satellite knot of winding number \(\neq 0\), then the unknotting number of K is greater than one.

Corollary: An unknotting number one knot is prime.

First, we show that there are knots for which a crossing change which lowers the unknotting number raises the genus. In fact the argument seems, in principle, to produce a myriad of such examples.

Second, we examine the effect of companion tori on crossing changes. We discover that either such companion tori are essentially unrelated to the part of the knot on which the crossing change is being made, or there is an inequality which relates the genus of the knot before and after the crossing change.

The precise statement and some corollaries are:

Definition: Let K be a knot in \(S^ 3\) and D be a disk which intersects K in precisely two points, of opposite orientation. Let \(L=\partial D\) be given a \(\pm 1\)-framing. Then L is a crossing link and D is its crossing disk. Denote by K’ the knot in \(S^ 3\) obtained from K by doing \(\pm 1\) surgery to L. (K and K’ are related by a single crossing change.) Let T be an incompressible torus in \(S^ 3-K.\)

Theorem: If genus(K’)\(\leq genus(K)-2\) then any companion torus of K can be isotoped disjoint from L.

Corollary: If K is a satellite knot of winding number \(\neq 0\), then the unknotting number of K is greater than one.

Corollary: An unknotting number one knot is prime.

### MSC:

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

### Keywords:

sutured manifold; crossing changes; genus of a knot; unknotting number; companion tori; crossing link; crossing disk; incompressible torus; satellite knot; winding number
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\textit{M. Scharlemann} and \textit{A. Thompson}, Math. Ann. 280, No. 2, 191--205 (1988; Zbl 0616.57003)

### References:

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[8] | Scharlemann, M.: Smooth spheres in ?4 with four critical points are standard. Invent. Math.79, 125-141 (1985) · Zbl 0559.57019 · doi:10.1007/BF01388659 |

[9] | Scharlemann, M.: Unknotting number one knots are prime. Invent. Math.82, 37-55 (1985) · Zbl 0576.57004 · doi:10.1007/BF01394778 |

[10] | Scharlemann, M.: Sutured manifolds and generalized Thurston norms. To appear in J. Differ. Geom. · Zbl 0673.57015 |

[11] | Thompson, A.: PropertyP for the band-connect sum of two knots. Topology26, 205-208 (1987) · Zbl 0628.57005 · doi:10.1016/0040-9383(87)90060-7 |

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