# Writhe

Writhe

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. If as you travel along a link component and cross over a crossing, the strand underneath goes from right to left, the crossing is positive; if the lower strand goes from left to right, the crossing is negative. One way of remembering this is to use a variation of the right-hand rule.

For a "knot" diagram, using the right-hand rule with either orientation gives the same result, so the writhe is well-defined on unoriented knot diagrams.

The writhe of a knot is unaffected by two of the three Reidemeister moves: moves of Type II and Type III do not affect the writhe. Reidemeister move Type I, however, increases or decreases the writhe by 1. This implies that the writhe of a knot is "not" an isotopy invariant of the knot itself &mdash; only the diagram. By a series of Type I moves one can set the writhe of a diagram for a given knot to be any integer at all.

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• Writhe — Writhe, v. i. To twist or contort the body; to be distorted; as, to writhe with agony. Also used figuratively. [1913 Webster] After every attempt, he felt that he had failed, and writhed with shame and vexation. Macaulay. [1913 Webster] …   The Collaborative International Dictionary of English

• writhe — writhe, agonize, squirm are comparable when they mean to twist or turn in physical or mental distress. Writhe regularly carries vivid suggestions of convulsive contortions (as of one in the throes of death, in a paroxysm, in an instrument of… …   New Dictionary of Synonyms

• Writhe — Writhe, v. t. [imp. {Writhed}; p. p. {Writhed}, Obs. or Poetic {Writhen}; p. pr. & vb. n. {Writhing}.] [OE. writhen, AS. wr[=i]?an to twist; akin to OHG. r[=i]dan, Icel. r[=i]?a, Sw. vrida, Dan. vride. Cf. {Wreathe}, {Wrest}, {Wroth}.] [1913… …   The Collaborative International Dictionary of English

• writhe — [raıð] v [: Old English; Origin: writhan [i] to twist ] to twist your body from side to side violently, especially because you are suffering pain writhe in pain/agony etc ▪ He lay writhing in pain …   Dictionary of contemporary English

• writhe — [ raıð ] verb intransitive to move by twisting and turning, especially when you feel a lot of pain: writhe in pain/agony: He writhed in agony on the ground …   Usage of the words and phrases in modern English

• writhe — index beat (pulsate), contort Burton s Legal Thesaurus. William C. Burton. 2006 …   Law dictionary

• writhe — (v.) O.E. wriðan to twist or bend, earlier to bind or fetter, from P.Gmc. *writhanan (Cf. N.Fris. wrial, O.H.G. ridan, O.N. riða, M.Swed. vriþa, M.Da. vride), from PIE *wreit to turn, bend (see WREATH (Cf. wreath)). Related …   Etymology dictionary

• writhe — [v] contort; toss back and forth agonize, bend, distort, jerk, recoil, squirm, struggle, suffer, thrash, thresh, twist, wiggle, wince, worm, wriggle; concepts 80,150 Ant. be still …   New thesaurus

• writhe — ► VERB ▪ twist or squirm in pain or as if in pain. ORIGIN Old English, «make into coils, plait» …   English terms dictionary

• writhe — [rīth] vt. writhed, writhing [ME writhen < OE writhan, to twist, wind about, akin to ON rītha < IE base * wer , to bend, twist > WREATH, WRY] to cause to twist or turn; contort vi. 1. to make twisting or turning movements; contort the… …   English World dictionary