Vortex theory inspired P.G. Tait to begin with the enumeration and classification of knotted structures and solve the census problem. For that he developed Scheme-method, a representation of reduced knot diagrams by codes (already known to Gauss) and the Partition method, an improvement of Listing's attempt. In collaboration with the Reverend T.P. Kirkman and C.N. Little, they succeeded in making a list of all alternating knots up to 11 crossings. The derivation of knots with 10 crossings took them six years to complete. In his works Tait also introduced and considered some of the fundamental problems in knot theory: chirality and unknotting number (or Gordian number, called by Tait "beknottedness"), and introduced the graph of a knot. He made several conjectures on alternating knots, e.g., that the minimal number of crossings of an alternating KL is always realized in its alternating diagram without loops, and that two minimal diagrams of the same oriented KL have the same writhe. His famous Flyping Conjecture was recently proved by Menasco and Thistlethwaite (1993), 100 years after the time it was formulated. Kirkman's geometrical system for the systematic derivation of knot projections (4-valent planar graphs) was closely related to the enumeration of basic polyhedra and, at the same time, represented a geometrical method for classifying knot projections. Kirkman derived the census of 1581 plane curves with 11 crossings from which Little distinguished 357 different alternating knots. Little also considered the derivation of non-alternating knots exponentially increasing the number of possibilities. In the case of non-alternating KLs, in addition to the flype, Little introduced a 2-pass: a KL transformation where a string is simply passed over a tangle. Working on it for six years, Little produced a catalogue consisting of 43 non-alternating knots with n = 10 crossings and 551 drawings of their various minimal projections (with few omissions). The only serious error in his tables was the duplication discovered by K. Perko in 1974. Little observed that the writhe of a reduced knot diagram is invariant with respect to flypes and 2-passes, and proposed that it is a knot invariant, but it is not: the first known counterexample is Perko pair. Little erroneously believed that just two kinds of moves, flypes and 2-passes, are sufficient to pass between all minimal diagrams of the same knot. In his derivation of non-alternating knots and in the knot minimization program knotfind.c (the part of Knotscape) M. Thistlethwaite used 13 different diagrammatic moves (Hoste, Thistlethwaite and Weeks, 1998). After Tait, Kirkman and Little almost nothing important happened for almost a century pertaining to knot tabulation until the works of J.H. Conway and A. Caudron in 1970-80s, and the computer derivation of KLs.
The principal problem in knot enumeration is to decide when two knots are ambient isotopic. Two KLs are the same or isotopy equivalent if one of them can be transformed to the other by pushing and pulling, but not cutting, its string(s). The problem of isotopy, known as the Knot problem, became the main problem in knot theory.
Closely connected to it is the problem of chirality, non-equality of a KL to its mirror image. Thirty years after Tait's first results in enumeration of achiral knots with n £10 crossings, M.G. Hasseman in his dissertation partially extended knot tables, and described achiral knots with n £ 12 crossings. Tait conjectured that every achiral KL must have an even number of crossings. Therefore neither Tait nor Haseman considered the possibility of the existence of achiral knots with an odd crossing number. The first oriented achiral link 8*.-2 0.2 0.-2 0 with n = 11 crossings was discovered in 1998 (Liang, Mislow and Flapan, 1998). The achiral non-alternating knot 10**2 0.2..-2 0..2 0.-1.-1.-1.-2 0 with n = 15 crossings was recently found by M. Thistlethwaite, who also recognized several duplications in Haseman's tables. However, the Tait Conjecture about achiral KLs holds for alternating KLs: there is no alternating achiral KL with an odd number of crossings.
The emphasis in the theory of knots turned away from enumeration toward attempts to prove the completeness of knot lists and to show that they do not contain repetitions. The first steps in the development of the required mathematical apparatus were made by H. Poincaré, who introduced several of topological objects and tools, e.g., the concept of the complex and its fundamental group. The first proof of the existence of non-trivial knots is given by H. Tietze in 1906. By using a more refined presentation of a fundamental group of knot, M. Dehn proved in 1914 that the oriented trefoil is not isotopic to its mirror image, i.e., that trefoil knot is chiral.