A full account is now available:

Infinite families of links with trivial Jones polynomial, by Shalom Eliahou, Louis H. Kauffman and Morwen Thistlethwaite (pdf, 330 Kb)

Added March 22, 2001:

We have found a common explanation for the sequences presented below, and are now able to prove:

Theorem.  Let L be any link with k >= 1 components; let V_L be the Jones polynomial of L, and let u denote the Jones polynomial of the unlink of two components, i.e. u = -t^-1/2 - t^1/2. Then there are infinitely many distinct prime links with (k+1) components whose Jones polynomial is equal to uV_L.

Corollary.  For each k >= 2 there are infinitely many prime k-component links with Jones polynomial equal to that of the k-component unlink.

Full details will appear shortly. Here are three examples:

A prime 4-component link with Jones polynomial u³

A prime 5-component link with Jones polynomial u⁴

A prime 6-component link with Jones polynomial u⁵

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

These links are all satellites of the Hopf link.

The accompanying pictures were produced with the help of Kenneth Stephenson's circle packing program.

(i) The family of 2-component links LL1(n) (n any integer).

For even n, the Jones polynomial of LL1(n) is precisely the Jones polynomial of the 2-component unlink, whereas for odd n the polynomial is equal to the unlink polynomial times t⁶ or t^-6, depending on choices of orientation.


LL1(n) is formed by clasping together the numerator of the rational tangle    n · 1 · 1 · 1 · 2    with the numerator of the rational tangle  -3 , as illustrated below.

For positive n the resulting diagram has n + 16 crossings, and for n <= 0 we get a diagram with |n| + 16 crossings reducible to |n| + 15 crossings, except that for n = -1 we can reduce to 14 crossings. Indeed, an exhaustive search has shown that LL1(-1), LL1(0) are the only links of up to 15 crossings with Jones polynomial equal to the unlink polynomial times a power of t.

LL1(1)

LL1(2)

LL1(3)

(ii) The family of 2-component links LL2(n) (n>=0).

The Kauffman bracket polynomial of LL2(n) is equal to that of the 2-component unlink for all n. However, the writhe is 0 for n even , and 8 (or -8) for n odd. Therefore, as with the previous family, the Jones polynomial of LL2(n) is equal to that of the 2-component unlink for all even n, but for odd n is equal to the unlink polynomial times a factor t⁶ or t^-6.

The link LL2(n) is formed similarly from the rational tangle    5 · 1 · 4 · 1 · 4 · 1 ·   ...   · 4 · 1 · 2    (where there are n occurrences of the pair  4 · 1 ) and its mirror-image, giving a diagram of 10n+24 crossings. Here are pictures of LL2(0) and LL2(1):

LL2(0)

LL2(1)

(iii) The family of 3-component links LLL(n) (n>=0).

The Jones polynomial of LLL(0) is equal to that of the 3-component unlink if one chooses orientations of the blue and green strands so that the writhe of the diagram is 2, whereas for n >= 1 the Jones polynomial of LLL(n) is equal to that of the 3-component unlink unconditionally.

This time we use the rational tangle  -3  together with the "generalized rational tangle"
2 · 1/2 · 1 · 1/2 · 1   ...   · 1/2 · 1 · 1 · 2    ( n occurrences of the pair  1/2 · 1 ) producing a diagram with 3n+16 crossings. Here are the first four links in this sequence:

LLL(0)

LLL(1)

LLL(2)

LLL(3)

Added March 5, 2001: The family of 2-component links LL1(m, n) (m>=0 , n any integer).

Further investigation has shown that the original family LL1(n) is the 1-parameter subfamily corresponding to m = 0 of this new 2-parameter family. LL1(m, n) is constructed from the rational tangle  -3 , together with
n · 1 · 1 · 1/2 · 1 · 1/2 · 1   ...   · 1/2 · 1 · 1 · 2    (m occurrences of the pair  1/2 · 1 ).

For m >= 1 the Jones polynomial of LL1(m, n) is precisely equal to that of the unlink of two components. Here are some examples: