The Proof

## Proving The Links Are Prime

Before one starts the appropriate calculating link invariants, one needs to note the following. First, if two links consist of A and of B number of components, their connected sum consists of A+B-1 and their unlinked union of A+B number of components. This is the reason why in the preceding page we were able to state that the knots need only be distinguished from the connected sum of knots, and to make similar statements for the "two-links" and the "three-links".

Second, if the linking numbers of any two links are x(1)-x(2)-...-x(n) and y(1)-y(2)-...-y(m), the linking numbers, both for their connected sum and for their unlinked union, are going to be x(1)-x(2)-...-x(n)-y(1)-y(2)-...-y(m)-0-0-...-0. Note that the number of 0's will be larger for the unlinked union, since it has one more component.

Third, if for a particular c-color colorization, two links' invariants are equal to A and B, the invariant for their connected sum is equal to A*B, and for their unlinked union is equal to A*B*c. Threfore the 118 links can be shown prime as follows.

KNOTS

Knot (3-1)#(3-1)

Invariant 2-1

Since the value for knot 3-1 is 3, the value for knot (3-1)#(3-1) is 9. The following knots have values other than 9.

3 | 3-1, 6-4, 7-1, 7-9, 8-4, 8-5, 8-6, 8-12, 8-13, 8-21, 8-27, 9-18, 9-23, 9-26, 9-31, 9-33, 9-39, 9-42, 9-46, 9-48, 9-50, 9-55, 9-56, 9-60, 9-62, 9-64, 9-66

1 | 4-1, 5-1, 5-3, 6-1, 6-3, 7-3, 7-4, 7-6, 7-8, 7-10, 8-1, 8-15, 8-17, 8-18, 8-19, 8-20, 8-22, 8-23, 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, 9-20, 9-24, 9-25, 9-28, 9-29, 9-30, 9-32, 9-34, 9-38, 9-40, 9-41, 9-44, 9-47, 9-53, 9-54, 9-57, 9-58, 9-59, 9-61, 9-63

Invariant 2-1-1

Since the value for knot 3-1 is 5, the value for knot (3-1)#(3-1) is 25. The following knots have values other than 25.

21 | 9-35, 9-36, 9-37, 9-65

17 | 8-24, 9-51

Knot (4-1)#(3-1)

Invariant 2-1

Since the value for knot 4-1 is 1 and for knot 3-1 is 3, the value for knot (4-1)#(3-1) is 3. The following knots have values other than 3.

9 | 8-24, 9-35, 9-36, 9-37, 9-51, 9-65

1 | 4-1, 5-1, 5-3, 6-1, 6-3, 7-3, 7-4, 7-6, 7-8, 7-10, 8-1, 8-15, 8-17, 8-18, 8-19, 8-20, 8-22, 8-23, 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, 9-20, 9-24, 9-25, 9-28, 9-29, 9-30, 9-32, 9-34, 9-38, 9-40, 9-41, 9-44, 9-47, 9-53, 9-54, 9-57, 9-58, 9-59, 9-61, 9-63

Invariant 2-1-1

Since the value for knot 4-1 is 1 and for knot 3-1 is 5, the value for knot (3-1)#(3-1) is 5. The following knots have values other than 25.

9 | 8-4, 8-5, 8-6, 8-12, 8-13, 9-18, 9-23, 9-46, 9-62

Invariant 3-1

Since the value for knot 4-1 is 4 and for knot 3-1 is 4, the value for knot (3-1)#(3-1) is 16. The following knots have values other than 16.

4 | 3-1, 8-21, 9-42, 9-55, 9-56, 9-66

1 | 6-4, 7-1, 7-9, 9-26, 9-31, 9-33, 9-39, 9-48, 9-50, 9-60, 9-64

The linking numbers of the connected sums (3-2)#(3-1) and (4-1)#(3-2) are 1, while the linking numbers of the unlinked unions (3-1)U(T/K), (3-1)U(3-1), (4-1)U(T/K), (5-1)U(T/K), (5-3)U(T/K), (6-1)U(T/K), (6-3)U(T/K) and (6-4)U(T/K) are 0. Therefore all two-component links with linking numbers 2, 3 and 4 have been shown prime.

We need only consider the links 3-2, 8-2, 8-16, 9-8 and 9-19. This is how they are distinguished from the connected sums (3-2)#(3-1) and (4-1)#(3-2).

3-2 and (3-2)#(3-1) Colorization 2-1: invariants are 1 and 3.

3-2 and (4-1)#(3-2) Colorization 3-1: invariants are 2 and 8.

8-2 and (3-2)#(3-1) Colorization 2-1-1: invariants are 1 and 10.

8-2 and (4-1)#(3-2) Colorization 2-1: invariants are 3 and 1.

8-16 and (3-2)#(3-1) Colorization 2-1: invariants are 1 and 3.

8-16 and (4-1)#(3-2) Colorization 3-1: invariants are 5 and 8.

9-8 and (3-2)#(3-1) Colorization 2-1: invariants are 1 and 3.

9-8 and (4-1)#(3-2) Colorization 2-2-1: invariants are 3 and 27.

9-19 and (3-2)#(3-1) Colorization 2-1-1: invariants are 6 and 10.

9-19 and (4-1)#(3-2) Colorization 3-1: invariants are 3 and 1.

We need only consider the links 6-2, 8-3, 8-8, 8-14, 8-25, 9-5, 9-7, 9-9 and 9-21. To distinguish them from the unlinked unions (3-1)U(T/K), (3-1)U(3-1), (4-1)U(T/K), (5-1)U(T/K), (5-3)U(T/K), (6-1)U(T/K), (6-3)U(T/K) and (6-4)U(T/K). For 8 out of the 9 links, this is shown from the fact that their 2-1 colorization invariant is equal to 1, while for all unlinked unions it is a multiple of 3. The only exception is link 8-25, whose 2-1 colorization invariant is equal to 3 (the number of 2-1 "colors"). This is how one proves that 8-25 is also prime.

The 2-1 invariants for the unlinked unions (3-1)U(T/K), (3-1)U(3-1), (4-1)U(T/K), (5-1)U(T/K), (5-3)U(T/K), (6-1)U(T/K), (6-3)U(T/K) and (6-4)U(T/K) are 3*1*3=9, 3*3*3=27, 1*1*3=3, 1*1*3=3, 1*1*3=3, 1*1*3=3, 1*1*3=3 and 3*1*3=9 respectively. Therefore 8-25 is distinct from (3-1)U(T/K), (3-1)U(3-1) and (6-4)U(T/K).

The 2-1-1 invariants for the remaining unlinked unions are all equal to 1*1*6=6, while the 2-1-1 invariant for 8-25 is 10. Therefore 8-25 is also shown prime.