The Jones Polynomial
(For In[1] see Setup)

In Naming and Enumeration we checked that the knots 6_1 and 9_46 have the same Alexander polynomial. Their Jones polynomials are different, though:
In[3]:=

Jones[Knot[6, 1]][q]

Out[3]=

4 3 2 2 2
2 + q  q + q    q + q
q

In[4]:=

Jones[Knot[9, 46]][q]

Out[4]=

6 5 4 2 2 1
2 + q  q + q   + q  
3 q
q

L8a6 
On links with an even number of components the Jones polynomial is a function of , and hence it is often more convenient to view it as a function of , where :
In[5]:=

Jones[Link[8, Alternating, 6]][q]

Out[5]=

(9/2) (7/2) 3 3 4 3/2
q + q   +    + 3 Sqrt[q]  2 q +
5/2 3/2 Sqrt[q]
q q
5/2 7/2
2 q  q

In[6]:=

PowerExpand[Jones[Link[8, Alternating, 6]][t^2]]

Out[6]=

9 7 3 3 4 3 5 7
t + t   +    + 3 t  2 t + 2 t  t
5 3 t
t t

The Jones polynomial attains 2110 values on the 2226 knots and links known to KnotTheory`
:
In[7]:=

all = Join[AllKnots[], AllLinks[]];

In[8]:=

Length /@ {Union[Jones[#][q]& /@ all], all}

Out[8]=

{2110, 2226}

How is the Jones polynomial computed?
(See also: The Kauffman Bracket using Haskell)
The Jones polynomial is so simple to compute using Mathematica that it's worthwhile pause and see how this is done, even for readers with limited prior programming experience. First, recall (say from [Kauffman]) the definition of the Jones polynomial using the Kauffman bracket :
[KBDef] 
here is a commutative variable, , and is the writhe of , the difference where and count the positive Failed to parse (unknown function "\overcrossing"): {\displaystyle (\overcrossing)} and negative Failed to parse (unknown function "\undercrossing"): {\displaystyle (\undercrossing)} crossings of respectively.
Just for concreteness, let us start by fixing to be the trefoil knot shown above:
In[9]:=

L = PD[Knot[3, 1]]

Out[9]=

PD[X[1, 4, 2, 5], X[3, 6, 4, 1], X[5, 2, 6, 3]]

Our first task is to perform the replacement Failed to parse (unknown function "\slashoverback"): {\displaystyle \langle\slashoverback\rangle\to A\langle\hsmoothing\rangle + B\langle\smoothing\rangle} on all crossings of . By our conventions (see Planar Diagrams) the edges around a crossing are labeled , , and : Failed to parse (unknown function "\slashoverback"): {\displaystyle {}^c_d\slashoverback{}_a^b} . Labeling the smoothings Failed to parse (unknown function "\hsmoothing"): {\displaystyle (\hsmoothing, \ \smoothing)} in the same way, Failed to parse (unknown function "\hsmoothing"): {\displaystyle {}^c_d\hsmoothing{}_a^b} and Failed to parse (unknown function "\smoothing"): {\displaystyle {}^c_d\smoothing{}_a^b} , we are lead to the symbolic replacement rule . Let us apply this rule to , switch to a multiplicative notation and expand:
In[10]:=

t1 = L /. X[a_,b_,c_,d_] :> A P[a,d] P[b,c] + B P[a,b] P[c,d]

Out[10]=

PD[A P[1, 5] P[2, 4] + B P[1, 4] P[2, 5],
B P[1, 4] P[3, 6] + A P[1, 3] P[4, 6],
A P[2, 6] P[3, 5] + B P[2, 5] P[3, 6]]

In[11]:=

t2 = Expand[Times @@ t1]

Out[11]=

2
A B P[1, 4] P[1, 5] P[2, 4] P[2, 6] P[3, 5] P[3, 6] +
2 2
A B P[1, 4] P[2, 5] P[2, 6] P[3, 5] P[3, 6] +
2 2
A B P[1, 4] P[1, 5] P[2, 4] P[2, 5] P[3, 6] +
3 2 2 2
B P[1, 4] P[2, 5] P[3, 6] +
3
A P[1, 3] P[1, 5] P[2, 4] P[2, 6] P[3, 5] P[4, 6] +
2
A B P[1, 3] P[1, 4] P[2, 5] P[2, 6] P[3, 5] P[4, 6] +
2
A B P[1, 3] P[1, 5] P[2, 4] P[2, 5] P[3, 6] P[4, 6] +
2 2
A B P[1, 3] P[1, 4] P[2, 5] P[3, 6] P[4, 6]

In the above expression the product P[1,4] P[1,5] P[2,4] P[2,6] P[3,5] P[3,6] represents a path in which 1 is connected to 4, 1 is connected to 5, 2 is connected to 4, etc. (see the right half of the figure above). We simplify such paths by repeatedly applying the rules and :
In[12]:=

t3 = t2 //. {P[a_,b_]P[b_,c_] :> P[a,c], P[a_,b_]^2 :> P[a,a]}

Out[12]=

3 2
B P[1, 1] P[2, 2] P[3, 3] + A B P[2, 2] P[4, 4] +
3 2 2
A P[3, 3] P[4, 4] + A B P[3, 3] P[4, 4] + 3 A B P[5, 5] +
2
A B P[1, 1] P[5, 5]

To complete the computation of the Kauffman bracket, all that remains is to replace closed cycles (paths of the form by , to replace by , and to simplify:
In[13]:=

t4 = Expand[t3 /. P[a_,a_] > A^2B^2 /. B > 1/A]

Out[13]=

9 1 3 7
A +  + A + A
A

We could have, of course, combined the above four lines to a single very short program, that compues the Kauffman bracket from the beginning to the end:
In[14]:=

KB0[pd_] := Expand[
Expand[Times @@ pd /. X[a_,b_,c_,d_] :> A P[a,d] P[b,c] + 1/A P[a,b] P[c,d]]
//. {P[a_,b_]P[b_,c_] :> P[a,c], P[a_,b_]^2 :> P[a,a], P[a_,a_] > A^21/A^2}]

In[15]:=

t4 = KB0[PD[Knot[3, 1]]]

Out[15]=

9 1 3 7
A +  + A + A
A

We will skip the uninteresting code for the computation of the writhe here; it is a linear time computation, and if that's all we ever wanted to compute, we wouldn't have bothered to purchase a computer. For our the result is , and hence the Jones polynomial of is given by
In[16]:=

(A^3)^(3) * t4 / (A^21/A^2) /. A > q^(1/4) // Simplify // Expand

Out[16]=

4 3 1
q + q + 
q

L11a548 
At merely 3 lines of code, our program is surely nice and elegant. But it is very slow:
In[17]:=

time0 = Timing[KB0[PD[Link[11, Alternating, 548]]]]

Out[17]=

23 5 10 3 5 13 17 21 25
{1., A +  +  + A + 6 A + 6 A + 5 A  5 A + 4 A  A }
15 7
A A

Here's the much faster alternative employed by KnotTheory`
:
In[18]:=

KB1[pd_PD] := KB1[pd, {}, 1];
KB1[pd_PD, inside_, web_] := Module[
{pos = First[Ordering[Length[Complement[List @@ #, inside]]& /@ pd]]},
pd[[pos]] /. X[a_,b_,c_,d_] :> KB1[
Delete[pd, pos],
Union[inside, {a,b,c,d}],
Expand[web*(A P[a,d] P[b,c]+1/A P[a,b] P[c,d])] //. {
P[e_,f_]P[f_,g_] :> P[e,g], P[e_,_]^2 :> P[e,e], P[e_,e_] > A^21/A^2
}
]
];
KB1[PD[],_,web_] := Expand[web]

In[19]:=

time1 = Timing[KB1[PD[Link[11, Alternating, 548]]]]

Out[19]=

23 5 10 3 5 13 17 21
{0.015, A +  +  + A + 6 A + 6 A + 5 A  5 A + 4 A 
15 7
A A
25
A }

(So on L11a548 KB1
is 23 5 10 3 5 13 17 21 25
{1., A +  +  + A + 6 A + 6 A + 5 A  5 A + 4 A  A }1,1
15 7 A A/ 23 5 10 3 5 13 17 21 25
{0.015, A +  +  + A + 6 A + 6 A + 5 A  5 A + 4 A  A }1,1
15 7 A A ~ 23 5 10 3 5 13 17 21 25 {1., A +  +  + A + 6 A + 6 A + 5 A  5 A + 4 A  A }1,1 15 7 A A
Round[]
23 5 10 3 5 13 17 21 25
{0.015, A +  +  + A + 6 A + 6 A + 5 A  5 A + 4 A  A }1,1
15 7
A A times faster than KB0
.)
The idea here is to maintain a "computation front", a planar domain which starts empty and gradualy increases until the whole link diagram is enclosed. Within the front, the rules defining the Kauffman bracket, Equation [KBDef], are applied and the result is expanded as much as possible. Outside of the front the link diagram remains untouched. At every step we choose a crossing outside the front with the most legs inside and "conquer" it  apply the rules of [KBDef] and expand again. As our new outpost is maximally connected to our old territory, the length of the boundary is increased in a minimal way, and hence the size of the "web" within our front remains as small as possible and thus quick to manipulate.
In further detail, the routine KB1[pd, inside, web]
computes the
Kauffman bracket assuming the labels of the edges inside the front are in
the variable inside
, the alreadycomputed inside of the front is in
the variable web
and the part of the link diagram yet untouched is
pd
. The single argument KB1[pd]
simply calls
KB1[pd, inside, web]
with an empty inside
and with web
set to 1. The three argument KB1[pd, inside, web]
finds the position of the crossing maximmally connected to the front using the somewhat
cryptic assignment
pos = First[Ordering[Length[Complement[List @@ #, inside]]& /@ pd]]}
KB1[pd, inside, web]
then recursively calls
itself with that crossing removed from pd}, with its legs
added to the
inside
, and with web
updated in accordance
with [KBDef]. Finally, when pd
is empty, the output is
simply the value of web
.
[Kauffman] ^ L. H. Kauffman, On knots, Princeton Univ. Press, Princeton, 1987.