Distinguishing Links

PROVING THAT THE LINKS OBTAINED ARE DISTINCT

We are now going to demonstrate that the links presented in this page are actually distinct. This is done by obtaining values of link invariants for the links, and by showing that for any two links, there exists at least one invariant for which these values are different. We are going to use the following invariants.


By applying the "number of components" invariant, one is able to separate the original 118 links into the following three groups.

One component 3-1, 4-1, 5-1, 5-3, 6-1, 6-3, 6-4, 7-1, 7-3, 7-4, 7-6, 7-8, 7-9, 7-10, 8-1, 8-4, 8-5, 8-6, 8-12, 8-13, 8-15, 8-17, 8-18, 8-19, 8-20, 8-21, 8-22, 8-23, 8-24, 8-27, 8-28, 8-29, 8-30, 8-31, 8-32, 9-1, 9-2, 9-3, 9-4, 9-11, 9-12, 9-14, 9-16, 9-18, 9-20, 9-23, 9-24, 9-25, 9-26, 9-28, 9-29, 9-30, 9-31, 9-32, 9-33, 9-34, 9-35, 9-36, 9-37, 9-38, 9-39, 9-40, 9-41, 9-42, 9-44, 9-46, 9-47, 9-48, 9-50, 9-51, 9-53, 9-54, 9-55, 9-56, 9-57, 9-58, 9-59, 9-60, 9-61, 9-62, 9-63, 9-64, 9-65, 9-66 (84 links)

Two components 3-2, 5-2, 6-2, 7-2, 7-5, 7-7, 8-2, 8-3, 8-7, 8-8, 8-9, 8-14, 8-16, 8-28, 9-5, 9-6, 9-7, 9-8, 9-9, 9-13, 9-15, 9-17, 9-19, 9-21, 9-22, 9-27, 9-43, 9-45, 9-49, 9-52 (30 links)

Three components 8-10, 8-11, 8-26, 9-10 (4 links)


Since linking numbers are meaningful only for links consisting of two or more components, the 84 links of the "one-component" class (i.e. knots), cannot be distinguished by this invariant. The 30 links of the "two-component" class are going to be further divided into the following five subclasses.

Linking Number 0 6-2, 8-3, 8-8, 8-14, 8-25, 9-5, 9-7, 9-9, 9-21 (9 links)

Linking Number 1 3-2, 8-2, 8-16, 9-8, 9-19 (5 links)

Linking Number 2 5-2, 7-2, 8-7, 8-9, 9-6, 9-13, 9-15, 9-17, 9-52 (9 links)

Linking Number 3 7-5, 7-7, 9-22, 9-27 (4 links)

Linking Number 4 9-43, 9-45, 9-49 (3 links)

Similarly, the 4 links of the "three-component" class are divided into the following two subclasses.

Linking Numbers 0-0-0 8-26 (1 link)

Linking Numbers 1-1-1 8-10, 8-11, 9-10 (3 links)

Up to this point, we have divided the 118 links into 8 classes. The second from last class consists of only one link, 8-26, which becomes the first link to be proven distinct to all other 117. This is the famous Borromean link, that consists of three components, while the linking number between any of two of these three components is 0. In fact, were one to remove any one of the three components, the remaining two components would yield the trivial 2-link. The Borromean link, however, is not trivial, as we shall show later.

In the following page we shall continue by calculating the colorization invariants of the other 117 links, in order to show that they are distinct.


Charilaos Aneziris

Copyright 2002

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