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Later knot tables took two approaches to resolving this: some just skipped one of the entries without renumbering, and others renumbered the later entries to remove the hole. The resulting ambiguity has continued to the present day, and has been further compounded by mistaken attempts to correct errors caused by this that were themselves incorrect.
7 4 knot, "endless knot" 8 18 knot, "carrick mat" 10 161 /10 162, known as the Perko pair; this was a single knot listed twice in Dale Rolfsen's knot table; the duplication was discovered by Kenneth Perko; 12n242/(−2,3,7) pretzel knot (p, q)-torus knot - a special kind of knot that lies on the surface of an unknotted torus in R 3
In knot theory, the writhe is a property of an oriented link diagram. The writhe is the total number of positive crossings minus the total number of negative crossings. A direction is assigned to the link at a point in each component and this direction is followed all the way around each component. For each crossing one comes across while traveling in this direction, if the strand un
A table of prime knots up to seven crossings. The knots are labeled with Alexander–Briggs notation. Traditionally, knots have been catalogued in terms of crossing number. Knot tables generally include only prime knots, and only one entry for a knot and its mirror image (even if they are different) (Hoste, Thistlethwaite & Weeks 1998).
Tait subsequently began listing unique knots in the belief that he was creating a table of elements. He formulated what is now known as the Tait conjectures on alternating knots. (The conjectures were proved in the 1990s.) Tait's knot tables were subsequently improved upon by C. N. Little and Thomas Kirkman. [1]: 6
Two classes of knots: torus knots and pretzel knots; Cinquefoil knot also known as a (5, 2) torus knot. Figure-eight knot (mathematics) the only 4-crossing knot; Granny knot (mathematics) and Square knot (mathematics) are a connected sum of two Trefoil knots; Perko pair, two entries in a knot table that were later shown to be identical.
In knot theory, prime knots are those knots that are indecomposable under the operation of knot sum. The prime knots with ten or fewer crossings are listed here for quick comparison of their properties and varied naming schemes.
The Perko pair gives a counterexample to a "theorem" claimed by Little in 1900 that the writhe of a reduced diagram of a knot is an invariant (see Tait conjectures), as the two diagrams for the pair have different writhes. In some later knot tables, the knots have been renumbered slightly (knots 10 163 to 10 166 are renumbered as 10 162 to 10 ...