The Knot 2000 (K2K) function GetMirrorImageKnot (webMathematica GetMirrorImageKnot) generates P-data of the mirror image of any KL. 

The first tables of knots were made experimentally by Tait, Kirkman and Little in the second part of XIX century (Tait, 1876/77a,b,c, 1883/84, 1884/85; Kirkman, 1885a,b; Little, 1885, 1890, 1892, 1900 ). The first book which contains knot tables is Knotentheorie by K. Reidemeister (1932). In those tables knots are classified according to their crossing number, every knot is represented by one minimal projection and numbered according to the place it occupies in the tables (in the Alexander-Briggs notation). The choice of minimal projections is, probably, random. The only order that can be recognized is that non-alternating knots are given after alternating ones. After K.Reidemeister, in the most of knot theory books, knot projections are just redrawn, sometimes turned upside down (e.g., 76), but almost never changed into some non-isomorphic projection of the same knot. The most influential book containing extensive tables of KLs and using already Conway seminal paper (1970) and Conway notation is D.Rolfsen's Knots and Links (1976). The only repetition in his tables was found by K.Perko (the famous Perko pair), who corrected also the Conway's eleven crossing tables where four knots were missed (Perko, 1974, 1982).

The KL notation that follows Alexander-Briggs and Reidemeister's concept we will call the classical notation of KLs, where a symbol of the form nij denotes ith knot or link with n crossings and j components, and for knots the upper index j = 1 is omitted. This classical notation that gives no information about any KL except its place in knot tables has been preserved till now in the most of knot theory books. For the first non-alternating knot diagram appearing in knot tables, and denoted in classical notation as 819, one of the possible Gauss codes for it and its mirror image will be   

{{8},{1,2,3,4,8,6,2,3,5,8,7,1,4,5,6,7}}

and 

{{8},{1,2,3,4,8,6,2,3,5,8,7,1,4,5,6,7}}

respectively. In the same way, for the first non-alternating link, denoted 633, Gauss codes could be 

{{1,2,3,4},{4,3,5,6},{2,5,6,1}}

and 

{{1,2,3,4},{4,3,5,6}{2,5,6,1}}.


PreviousContentsNext