Involutions of knots that fix unknotting tunnels. (English) Zbl 1147.57004
Summary: Let \(K\) be a knot that has an unknotting tunnel \(\tau\). We prove that \(K\) admits a strong involution that fixes \(\tau\) pointwise if and only if \(K\) is a two-bridge knot and \(\tau\) its upper or lower tunnel.
MSC:
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
| 57M60 | Group actions on manifolds and cell complexes in low dimensions |
References:
| [1] | DOI: 10.1007/BF01444492 · Zbl 0830.57009 · doi:10.1007/BF01444492 |
| [2] | DOI: 10.1007/BF01456837 · Zbl 0627.57007 · doi:10.1007/BF01456837 |
| [3] | Burde G., Knots (1985) |
| [4] | DOI: 10.1007/BF02566211 · Zbl 0469.57005 · doi:10.1007/BF02566211 |
| [5] | DOI: 10.1016/S0079-8169(08)61637-2 · doi:10.1016/S0079-8169(08)61637-2 |
| [6] | DOI: 10.1007/BF01446565 · Zbl 0697.57002 · doi:10.1007/BF01446565 |
| [7] | Scharlemann M., J. Differential Geom. 34 pp 539– |
| [8] | Schubert H., Math. Z. 66 pp 133– |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
