Who is thy Neighbor?

When performing Reidemeister moves on a link notation, one needs to know which numbers are considered as "neighbors". The same need will arise when checking if two notations yield identical presentations. In the case of knots, this issue was quite simple: every number i was the neighbor of i-1 and i+1, unless i=1 (neighbors: 2 and 2N), or i=2N (neighbors: 2N-1 and 1). In the case of links, this statement has to be modified as follows.

If a number i corresponds to an artificial crossing, it has no neighbors. If it corresponds to a real crossing: if i+1 and / or i-1 correspond to real crossings, they are indeed its neighbors. If not, one proceeds as follows.

Let a "chain" or series of numbers: ...,x,y,z,w,... that has the following properties: if a number t corresponds to a real crossing, it is followed by t+1 (or 1 if t=2N). If t corresponds to an artificial crossing, we check whether its predecessor coincides with its notation "partner". If so, then t is also followed by t+1. If not, then t is followed by its notation "partner". In the example of a preceding page, notation {(1,4),(3,6),<5,2>}, we may define two such chains: 1-2-5-6-1 and 3-4-5-2-3.

In each such chain, a number corresponding to a real crossing is considered to have as "neighbors" the nearest numbers that also correspond to real crossings. In the example above, 1 has as "neighbor" the number 6 (from both sides), while 3 has 4 as its neighbor. 2 and 5 have no neighbors (correspond to the artificial crossing <2,5>).

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Charilaos Aneziris

Copyright 2001


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