Proving that the links are Non-Trivial

In order for a link to be included in the list, it is not sufficient to show that it is distinct from all other links in the list. One also needs to show the following. First, that the link is non-trivial. This means that the link cannot be continuously deformed to any of the links that is shown in this figure. Second, that it cannot be decomposed to a "combination" of simpler links. Links may be combined in two ways, as shown here. In this page we demonstrate the non-triviality of the 118 links of crossing number up to 9.

One starts by noticing that for the one-component trivial link (i.e. the trivial knot = {}, all colorization invariants are equal to 1. Therefore in order to show that a knot is non-trivial, one needs to obtain a colorization for which the invariant is larger than 1 (i.e. non-trivial colorings are permitted).

For multicomponent trivial links, one notices that all linking numbers are 0, while the colorization invariants are equal to c raised to the n-1 power, where c is the number of "colors" (distinct elements of the corresponding conjugacy class) and n the number of components. Therefore a link may be shown non-trivial if any of its linking numbers is non-zero, or if any of its colorization invariants is either larger or smaller than c to the n-1 power.

 LINK INVARIANT VALUE ---------- -------------------- --------- 3-1 2-1 3 4-1 3-1 4 5-1 3-2 7 5-3 2-2-1 9 6-1 3-1-1 7 6-3 3-2 7 6-4 2-1 3 7-1 2-1 3 7-3 4-1 13 7-4 4-2 17 7-6 3-1 4 7-8 3-1 4 7-9 2-1 3 7-10 2-2-2-1 49 8-1 3-2-1 13 8-4 2-1 3 8-5 2-1 3 8-6 2-1 3 8-12 2-1 3 8-13 2-1 3 8-15 2-2-1 9 8-17 3-2-2 25 8-18 3-2 7 8-19 3-1 4 8-20 3-2 7 8-21 2-1 3 8-22 3-1-1 7 8-23 3-1 4 8-24 2-1 9 8-27 2-1 3 8-28 3-1-1 7 8-29 2-2-1 9 8-30 2-2-1 9 8-31 3-1 4 8-32 5 11 9-1 3-1 4 9-2 3-1 4 9-3 3-1 4 9-4 3-1 4 9-11 3-1 4 9-12 3-1 4 9-14 4-2-1 17 9-16 3-1 4 9-18 2-1 3 9-20 4-1 17 9-23 2-1 3 9-24 3-1-1 7 9-25 3-1 4 9-26 2-1 3 9-28 3-2 7 9-29 4-1 9 9-30 2-2-1 9 9-31 2-1 3 9-32 3-2 7 9-33 2-1 3 9-34 3-1 4 9-35 2-1 9 9-36 2-1 9 9-37 2-1 9 9-38 5 6 9-39 2-1 3 9-40 3-1 4 9-41 3-1 4 9-42 2-1 3 9-44 4-1 9 9-46 2-1 3 9-47 4-1 5 9-48 2-1 3 9-50 2-1 3 9-51 2-1 9 9-53 3-1 4 9-54 3-1 4 9-55 2-1 3 9-56 2-1 3 9-57 3-1 4 9-58 4-2-1 33 9-59 3-1 4 9-60 2-1 3 9-61 5 6 9-62 2-1 3 9-63 3-2 13 9-64 2-1 3 9-65 2-1 9 9-66 2-1 3

We continue with the proof of the non-triviality of the 30 two-component links. Since the linking number of the trivial 2-link is 0, we need only be concerned with the 9 links whose linking number is 0. In order to be shown non-trivial, they must possess an invariant that is not equal to the number of "colors". These links are shown non-trivial as follows.

 LINK INVARIANT NUMBER OF COLORS VALUE ---------- -------------------- ------------------------------ --------- 6-2 2-1 3 1 8-3 2-1 3 1 8-8 2-1 3 1 8-14 2-1 3 1 8-25 2-1-1 6 10 9-5 2-1 3 1 9-7 2-1 3 1 9-9 2-1 3 1 9-21 2-1 3 1

We conclude this discussion with proving the non-triviality of the 4 three-component links (8-10, 8-11, 8-26 and 9-10). 3 out of thes 4 links have non-zero linking numbers. We thus need only be concerned with 8-26. Its non-triviality is shown through the invariant corresponding to the conjugacy class 2-1. While the three-component trivial link allows for 3**2 = 9 distinct "colorings" (assuming one fixes the "color" of one strand), 8-26 allows for only 1 "coloring" (where all 6 strands are mapped to identical "colors").

We have thus shown that none of the 118 listed links is trivial. In the following page we are going to demonstrate that none of them is equivalent to a "combination" of simpler links.

Charilaos Aneziris
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