Unknotting number

The unknotting number of a knot is the minimal number of crossing changes required to convert a knot into the unknot. Literature on the unknotting number is extensive.

Many people contributed to the list of values of unknotting numbers presented here. References are given when published for individual knots.

Special mention should go to Slavik Jablan and Radmila Sazdanovic for the values of the unknotting numbers of 11 crossing knots. They did the initial calculations, developing lower bounds based on the the nontriviality of the knots and the signature. Upper bounds were found using explicit calculations.

The lower bounds obtained by Jablan and Sazdanovic were found using the signature of the knot and the fact that these knots are nontrivial. According to the Bernhard-Jablan Conjecture (eg. see A.Stoimenow, On unknotting numbers and knot trivadjacency, Mathematica Scandinavica 94(2) (2004), 227-248. available on-line from the address: http://www.kurims.kyoto-u.ac.jp/~stoimeno/papers.html) a minimal sequence of unknotting crossing changes can be found from a sequence determined by particular diagrams, thus offering a conjectural algorithm for determining the unknotting number. The algorithm does produce an unknotting sequence, and thus does give an upper bound on the unknotting number, regardless of the truth of the conjecture.

Further information on particular knots.