**Infinite families of links with
trivial Jones polynomial**

Shalom Eliahou, Louis H. Kauffman and Morwen Thistlethwaite

*February 1, 2001*

## Added June 8, 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 u^{3}

A prime 5-component link
with Jones polynomial u^{4}

A prime 6-component link
with Jones polynomial u^{5}

#
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

### 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 t^{6} 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
t^{6} 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:

###
LL1(1, 1)

LL1(1, 2)

LL1(1, 3)

LL1(2, 1)

LL1(2, 2)

LL1(2, 3)