An invertible knot is a knot that can be deformed via an ambient isotopy into itself but with the orientation reversed. A knot that is not invertible is said to be noninvertible.
Knots with ten or fewer crossings can be tested in the Wolfram Language for invertibility using KnotData[knot, "Invertible"].
Fox (1962, p. 143) conjectured that the knot 
is noninvertible. Trotter (1963) proved that noninvertible
knots exist by constructing an infinite family, the smallest
of which has 15 crossings.
Three prime knots with 9 or fewer crossings are noninvertible: 
,
, and 
(Cromwell 2004, pp. 297-299). Some noninvertible
knots can be obtained in the Wolfram
Language as KnotData["Noninvertible"].
The simplest noninvertible knot, 
, is illustrated above (Whitten 1972).
The following table gives the numbers of noninvertible and invertible prime knots with knot crossing number 
, 2, 3, ..., counting each knot
together with its mirror image.
| type | OEIS | counts |
| noninvertible | A052403 | 0, 0, 0, 0, 0, 0, 0, 1, 2, 33, 187, 1144, 6919, 38118, 226581, 1309875, ... |
| invertible | A052402 | 0, 0, 1, 1, 2, 3, 7, 20, 47, 132, 365, 1032, 3069, 8854, 26712, 78830, ... |
Invertibility can be decided by testing equivalence of an oriented knot with the same knot carrying the opposite orientation. Coward and Lackenby (2014, Theorem 1.1) give an explicit bound on the number of Reidemeister moves needed between equivalent oriented knot diagrams, yielding an algorithm for this test. Burde and Zieschang (2002) give a tabulation from which the noninvertible knots with up to 10 crossings can be extracted.
Knots." Math. Intell. 20, 33-48,
Fall 1998.Sloane, N. J. A. Sequences