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Shorthands denote
u_a algebraic unknotting number
Nak Nakanishi index
det determinant
sign signature
max LT maximum absolute value of
Levine-Tristram signatures
Hidden features
Click on to see
algebraic unknotting number how it has been detected
Alexander polynomial a Seifert matrix
(nondegenerate representative in the S-equivalence class)
Nakanishi index generator of the Alexander module,
if Nakanishi index is 1
Determinant H_1 of the double branched cover


Welcome to the
KNOTORIOUS
world wide web page!
set up by Maciej Borodzik mcboro'at'mimuw;edu;pl
and Stefan Friedl sfriedl'at'gmail;com
last update of the webpage 19 Feb 2012
last update of the knotorious data 01 Dec 2011
You may freely contact the authors in case of any questions.

Criterion # of cases it detects u_a.
the signature 883 (30%)
the span of the Levine--Tristram signature 888 (30%)
the number of generators of H_1(double cover) 170 (6%)
higher covers 199 (7%)
the Lickorish obstruction 435 (15%)
the Stoimenow criterion 4 (0.1%)
the new u(K)=2 obstruction 32 (1%)
Total number of knots in the database: 2977

Remarks:

We want to show, how often the given invariant is the unique criterion among others. We restrict ourselves to the knots, for which we were able to show that u_a>1. The precise meaning of each entry is discussed below.

Criterion How often is it unique
(1) H_1(double cover) 125 (8%)
(2) higher covers 44 (3%)
(3) signature 815 (53%)
(4) the span of signatures 1 (0.1%)
(5) the Stoimenow criterion 4 (0.3%)
(6) the Lickorish obstruction 371 (24%)
(7) the new u(K)=2 obstruction 21 (1%)
Number of knots with unknotting 2 or more: 1526

Explanation.
  1. The entry H_1 of double cover counts all the knots for which the number of generators of H_1(double branched cover) is equal to the algebraic unknotting number, but the invariants in points (3),(4),(6) and (7) are insufficient to detect the unknotting number.
  2. The entry higher covers counts all the knots for which the number of generators of H_1(branched cover of order 3 or more) is equal to the algebraic unknotting, but the invariants in points (1),(3),(4),(5),(6),(7) are insufficient.
  3. The entry signature counts all the knots for which |signature(K)/2|=u_a(K), but all the invariants in points (1),(2),(5),(6),(7) cannot detect the unknotting number.
  4. The entry the span of signatures shows the number of the knots for which the invariants in points (1),(2),(3),(5),(6) and (7) do not detect the algebraic unknotting number, but u_a is equal to half the difference between the maximal and minimal value of the Levine--Tristram signature.
  5. The entry the Stoimenow criterion shows the number of knots, for which the Stoimenow's criterion shows that u_a(K)=3, but no invariants from (1),(2),(3),(4),(6) can give that result.
  6. The entry the Lickorish obstruction shows the number of all knots, for which the Lickorish obstruction shows that u_a=2, but no invariant from (1),(2),(3),(4),(5),(7) can show that result.
  7. The entry the new u(K)=2 obstruction shows the number of all knots, for which the new unknotting=2 obstruction detect that u_a(K)=3, but no criterion mentioned in points (1),(2),(3),(4),(5) and (6) is capable of doing that.
  8. The number of knots with unknotting 2 or more does not count 25 knots for which we do not know the unknotting number.