In every crossing point of a knot it is possible to make a crossing change: transform overcrossing to undercrossing or vice versa. The unknotting number u(k) of a knot k is the smallest number of such changes required to obtain the unknot, the minimum taken over all regular projections. According to the classical definition, we could after each change do an ambient isotropy, then perform next change in the new projection, etc., and continue in this manner until the unknot is obtained. According to the standard definition, we must perform all changes in a single (fixed) projection of k. These two definitions are equivalent [10, pp. 58]. |
If in the first definition we restrict "all the regular projections" to "all the minimal regular projections", we cannot always obtain the correct unknoting number. This shows the well known example of the knot 10_{8} (or 514 in Conway notation), given by Y. Nakanishi [32] and S. Bleiler [33]: the alternating knot 10_{8} with u(10_{8})=2 and with the only one minimal projection needing at least three crossing changes to be transformed to the unknot. The same property holds for the complete family (2m+1)1(2m) [34]. Let us reconsider this example, with regard to the traditional definition. The first change in the minimal projection can result in the three subknots: 8_{2} occuring four times, 8_{4} occuring five times, and 6_{2} occuring once. Their unknoting numbers are: u(8_{2})=2, u(8_{4})=2, u(6_{2})=1. So, the optimal strategy is to transform the projection of 6_{2} obtained from 10_{8} by the first change to the minimal projection, and then derive from it the unknot by the second change. The denoted central point is the unique point of 6_{2} for the change resulting in the unknot. If we accept the first definition, this cannot be done, simply because this point is obtained by the reduction of the projection after the first change, and not exists in the minimal projection of 10_{8}. This point of 6_{2} is introduced in the non-minimal projection of 10_{8}, giving the correct uncrossing number u(10_{8})=2. Therefore, we propose the following
Conjecture:
(b) u(k) = min u(k^{-})+1,
where the minimum is taken over all knots k^{-}, obtained
from a minimal projection of k by one change of crossing.
The Conjecture holds for all the exactly determined unknotting numbers (n £ 10) from the book "A Survey of Knot Theory" by A.Kawaguchi, and if in all the other cases (A=1 or 2; B=2 or 3, etc.) for the unknotting numbers we take the maximal values for A, B, etc. If any of that unknotting numbers is smaller than its maximal estimated value, this will be the counterexample for the Conjecture. This Conjecture, introduced by J.A.Bernhard in the paper "Unknotting numbers and their minimal knot diagrams", J. Knot Theory & Ramifications, 3, 1 (1994), 1-5, was also independently proposed by the author in 1995, and effectively used for the calculation of unknotting numbers of the knots with n £ 10 crossings. The unknotting number is the property of family. For example, for the knots p2, p11p, (p+2)11p, p212, p1112, u(k) = 1. By using Conjecture, it is possible to calculate unknotting numbers for different very large classes of knots, e.g. for all the knots of the form pq.
In the similar way it is possible to define ¥-crossing change,
transforming a crossing into uncrossing and preserving the number
of components. Such ¥-change is considered
in Mirror Curves
(Tripod Server, USA)
as the introduction of the corresponding mirror.
Every ¥-crossing change transforms an
alternating knot into other alternating knot. For
¥-unknotting numbers
u^{¥}(k)
we could formulate the analogous Conjecture. For n £ 10, u^{¥}(k) = 1
for linear knots 2p+1, u^{¥}(k)= 2
for the knots
pq, p1q (p+q =1
(mod 2)), p,q,r (p+q+r = 0
(mod 2)), and u(k) ³
3 for all the other knots.
For the knots p1q (p³q³2) from this table, the unknotting numbers are given in left upper corner. For all of them holds the relationship u(p1q) = min ((u((p-2)1q), u(p1(q-2)), u((p-2)1(q-2)) + 1, except for the knots of the family (2m+1)1(2m), where u((2m+1)1(2m)) = min ((u((2m-1)1(2m)), u((2m+1)1(2m-2)), u((2m)1(2m-1)) + 1. For all knots p1q, u^{¥} = 3 iff p + q = 0 (mod 2); otherwise, u^{¥} = 2. For every knot is given its Alexander polynomial, and -a^{k} is a shorthand for -a+a..., where a occurs k times. The recent progress is made by A.Stiomenow, who succeeded to prove that the Conjecture holds for a restricted class of knots: a rational knot of unknotting number one has an unknotting number one minimal diagram (A.Stoimenow: Vassiliev Invariants and Rational Knots of Unknotting Number One). |