tunnel number
In mathematics, the tunnel number of a knot, as first defined by Bradd Clark, is a knot invariant, given by the minimal number of arcs (called tunnels) that must be added to the knot so that the complement becomes a handlebody. The tunnel number can equally be defined for links. The boundary of a regular neighbourhood of the union of the link and its tunnels forms a Heegaard splitting of the link exterior.
Examples
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- The unknot is the only knot with tunnel number 0.
- The trefoil knot has tunnel number 1. In general, any nontrivial torus knot has tunnel number 1.{{cite journal |last1=Boileau |first1=Michel |last2=Rost |first2=Markus |last3=Zieschang |first3=Heiner |title=On Heegaard decompositions of torus knot exteriors and related Seifert fibre spaces |journal=Mathematische Annalen |date=1 January 1988 |volume=279 |issue=3 |pages=553–581 |doi=10.1007/BF01456287 |language=en |issn=1432-1807}}
Every link L has a tunnel number. This can be seen, for example, by adding a 'vertical' tunnel at every crossing in a diagram of L. It follows from this construction that the tunnel number of a knot is always less than or equal to its crossing number.
References
- {{citation
| last = Clark | first = Bradd
| title = The Heegaard Genus Of Manifolds Obtained By Surgery On Links And Knots
| journal = International Journal of Mathematics and Mathematical Sciences
| volume = 3
| issue = 3
| pages = 583–589
| year = 1980
| doi = 10.1155/S0161171280000440| doi-access = free
}}
- {{citation
| last1 = Boileau | first1 = Michel
| last2 = Lustig | first2 = Martin
| last3 = Moriah | first3 = Yoav
| doi = 10.1017/S0305004100071930
| issue = 1
| journal = Mathematical Proceedings of the Cambridge Philosophical Society
| mr = 1253284
| pages = 85–95
| title = Links with super-additive tunnel number
| volume = 115
| year = 1994| bibcode = 1994MPCPS.115...85B}}.
- {{citation
| last1 = Kobayashi | first1 = Tsuyoshi
| last2 = Rieck | first2 = Yo'av
| doi = 10.1515/CRELLE.2006.023
| journal = Journal für die reine und angewandte Mathematik
| mr = 2222730
| pages = 63–78
| title = On the growth rate of the tunnel number of knots
| volume = 2006
| year = 2006| issue = 592
| arxiv = math/0402025}}.
- {{citation
| last = Scharlemann | first = Martin
| doi = 10.1016/0166-8641(84)90013-0
| issue = 2–3
| journal = Topology and Its Applications
| mr = 769294
| pages = 235–258
| title = Tunnel number one knots satisfy the Poenaru conjecture
| volume = 18
| year = 1984| doi-access = free
}}.
- {{citation
| last = Scharlemann | first = Martin
| doi = 10.1090/S0002-9947-03-03182-9 | doi-access=free
| issue = 4
| journal = Transactions of the American Mathematical Society
| mr = 2034312
| pages = 1385–1442
| title = There are no unexpected tunnel number one knots of genus one
| volume = 356
| year = 2004| arxiv = math/0106017
}}.
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