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

**TWO-COMPONENT LINKS**

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.

__Linking Number 1__

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

__ 3-2 and (4-1)#(3-2)__ Colorization

__ 8-2 and (3-2)#(3-1)__ Colorization

__ 8-2 and (4-1)#(3-2)__ Colorization

__ 8-16 and (3-2)#(3-1)__ Colorization

__ 8-16 and (4-1)#(3-2)__ Colorization

__ 9-8 and (3-2)#(3-1)__ Colorization

__ 9-8 and (4-1)#(3-2)__ Colorization

__ 9-19 and (3-2)#(3-1)__ Colorization

__ 9-19 and (4-1)#(3-2)__ Colorization

__Linking Number 0__

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.

**THREE-COMPONENT LINKS**

The linking numbers of (3-2)#(3-2), (3-2)U(T/K), (3-2)U(3-1), (5-2)U(T/K), (6-2)U(T/K) and (3-1)U(T/K)U(T/K) come out to be 1-1-0, 1-0-0, 1-0-0, 1-0-0, 0-0-0 and 0-0-0. Therefore, we need only consider the link 8-26, and the unlinked unions (5-2)U(T/K) and (6-2)U(T/K). 8-26 can be shown distinct from both unlinked unions from the fact that the 2-1 invariant of 8-26 is 1, while the 2-1 invariants of the unlinked unions are multiples of 3.

Charilaos Aneziris

**Copyright 2002**

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