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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.
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 ...
The directional figure eight (a.k.a. inline figure-eight loop) is a loop knot. It is a knot that can be made on the bight. The loop must only be loaded in the correct direction or the knot may fail. It is useful on a hauling line to create loops that can be used as handholds.
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.
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.
Trefoil knot without 3-fold symmetry with crossings labeled. A table of all prime knots with seven crossing numbers or fewer (not including mirror images).. In the mathematical area of knot theory, the crossing number of a knot is the smallest number of crossings of any diagram of the knot.
The unknotting number of a knot is always less than half of its crossing number. [2] This invariant was first defined by Hilmar Wendt in 1936. [3] Any composite knot has unknotting number at least two, and therefore every knot with unknotting number one is a prime knot. The following table show the unknotting numbers for the first few knots:
This is a typical element of the braid group, which is used in the mathematical field of knot theory. In mathematics Alexander's theorem states that every knot or link can be represented as a closed braid; that is, a braid in which the corresponding ends of the strings are connected in pairs.