The second knot polynomial discovered. Unlike the first-discovered Alexander polynomial,
the Jones polynomial can sometimes distinguish handedness (as can its more powerful
generalization, the HOMFLY polynomial). Jones
polynomials are Laurent polynomials in 
assigned to an 
knot. The Jones polynomials are
denoted 
for links, 
for knots, and normalized so
that
 |
(1)
|
For example, the right-hand and left-hand trefoil knots have polynomials
 |  |  |
(2)
|
 |  |  |
(3)
|
respectively.
If a link has an odd number of components, then 
is a Laurent polynomial over the integers;
if the number of components is even, 
is 
times a Laurent polynomial.
The Jones polynomial of a knot sum 
satisfies
 |
(4)
|
The skein relationship for under- and overcrossings is
 |
(5)
|
Combined with the link sum relationship, this allows Jones polynomials to be built up from simple knots and links to more complicated ones.
Some interesting identities from Jones (1985) follow. For any link 
,
 |
(6)
|
where 
is the Alexander polynomial, and
 |
(7)
|
where 
is the number of components of 
. For any knot 
,
 |
(8)
|
and
 |
(9)
|
Let 
denote the mirror image of a knot 
. Then
 |
(10)
|
Jones defined a simplified trace invariant for knots by
 |
(11)
|
The Arf invariant of 
is given by
 |
(12)
|
(Jones 1985), where i is 
. A table of the 
polynomials is given by Jones (1985) for knots of up to eight
crossings, and by Jones (1987) for knots of up to 10 crossings. (Note that in these
papers, an additional polynomial which Jones calls 
is also tabulated, but it is not the conventionally defined
Jones polynomial.)
Jones polynomials were subsequently generalized to the two-variable HOMFLY polynomials, the relationship being
 |
(13)
|
 |
(14)
|
They are related to the Kauffman polynomial F by
 |
(15)
|
Jones (1987) gives a table of braid words and 
polynomials for knots up to 10 crossings.
Jones polynomials for knots up to nine crossings are given
in Adams (1994) and for oriented links up to nine crossings by Doll and Hoste (1991).
All prime knots with 9 or fewer crossings have distinct
Jones polynomials. However, there exist distinct knots (and
even knots having different crossing numbers) that share the same Jones polynomial.
Examples include (05-001, 10-132), (08-008, 10-129), (08-016, 10-156), (10-025, 10-056),
(10-022, 10-035), (10-041, 10-094), (10-043, 10-091), (10-059, 10-106), (10-060,
10-083), (10-071, 10-104), (10-073, 10-086), (10-081, 10-109), and (10-137, 10-155)
(Jones 1987). (Incidentally, the first four of these also have the same HOMFLY
polynomial.)
It is not known if there is a nontrivial knot with Jones polynomial 1.
The Jones polynomial of an 
-torus knot is
 |
(16)
|
Let 
be one component of an oriented link 
. Now form a new oriented link 
by reversing the orientation of 
. Then
 |
(17)
|
where 
is the Jones polynomial and 
is the linking number
of 
and 
.
No such result is known for HOMFLY polynomials
(Lickorish and Millett 1988).
Birman and Lin (1993) showed that substituting the power series for 
as the variable in the Jones polynomial yields a power
series whose coefficients are Vassiliev
invariants.
Let 
be an oriented connected link projection of 
crossings, then
 |
(18)
|
with equality if 
is alternating and has no reducible
crossing (Lickorish and Millett 1988).
Witten (1989) gave a heuristic definition in terms of a topological quantum field theory, and Sawin (1996) showed that the "quantum group" 
gives rise to the Jones polynomial.
