The Jones Polynomial: Difference between revisions

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{{Startup Note}}
{{Startup Note}}



<!--$$?Jones$$-->
<!--$$?Jones$$-->
<!--Robot Land, no human edits to "END"-->
{{HelpLine|
n = 2 |
in = <nowiki>Jones</nowiki> |
out= <nowiki>Jones[L][q] computes the Jones polynomial of a knot or link L as a function of the variable q.</nowiki>}}
<!--END-->


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


<!--$$Jones[Knot[6, 1]][q]$$-->
<!--$$Jones[Knot[6, 1]][q]$$-->
<!--Robot Land, no human edits to "END"-->
{{InOut|
n = 3 |
in = <nowiki>Jones[Knot[6, 1]][q]</nowiki> |
out= <nowiki> -4 -3 -2 2 2
2 + q - q + q - - - q + q
q</nowiki>}}
<!--END-->
<!--END-->


<!--$$Jones[Knot[9, 46]][q]$$-->
<!--$$Jones[Knot[9, 46]][q]$$-->
<!--Robot Land, no human edits to "END"-->
{{InOut|
n = 4 |
in = <nowiki>Jones[Knot[9, 46]][q]</nowiki> |
out= <nowiki> -6 -5 -4 2 -2 1
2 + q - q + q - -- + q - -
3 q
q</nowiki>}}
<!--END-->
<!--END-->


{{Knot Image|L8a6|gif}}
The Jones polynomial attains <!--$all=Join[AllKnots[], AllLinks[]]; Length[Union[Jones[#][q]& /@ all]]$--><!--END--> values on the <!--$Length[all]$--><!--END--> knots and links known to <code>KnotTheory`</code>:

On links with an even number of components the Jones polynomial is a function of <math>\sqrt{q}</math>, and hence it is often more convenient to view it as a function of <math>t</math>, where <math>t^2=q</math>:

<!--$$Jones[Link[8, Alternating, 6]][q]$$-->
<!--Robot Land, no human edits to "END"-->
{{InOut|
n = 5 |
in = <nowiki>Jones[Link[8, Alternating, 6]][q]</nowiki> |
out= <nowiki> -(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</nowiki>}}
<!--END-->

<!--$$PowerExpand[Jones[Link[8, Alternating, 6]][t^2]]$$-->
<!--Robot Land, no human edits to "END"-->
{{InOut|
n = 6 |
in = <nowiki>PowerExpand[Jones[Link[8, Alternating, 6]][t^2]]</nowiki> |
out= <nowiki> -9 -7 3 3 4 3 5 7
-t + t - -- + -- - - + 3 t - 2 t + 2 t - t
5 3 t
t t</nowiki>}}
<!--END-->

The Jones polynomial attains <!--$all=Join[AllKnots[], AllLinks[]]; Length[Union[Jones[#][q]& /@ all]]$--><!--Robot Land, no human edits to "END"-->2110<!--END--> values on the <!--$Length[all]$--><!--Robot Land, no human edits to "END"-->2226<!--END--> knots and links known to <code>KnotTheory`</code>:


<!--$$all = Join[AllKnots[], AllLinks[]];$$-->
<!--$$all = Join[AllKnots[], AllLinks[]];$$-->
<!--Robot Land, no human edits to "END"-->
{{In|
n = 7 |
in = <nowiki>all = Join[AllKnots[], AllLinks[]];</nowiki>}}
<!--END-->
<!--END-->


<!--$$Length /@ {Union[Jones[#][q]& /@ all], all}$$-->
<!--$$Length /@ {Union[Jones[#][q]& /@ all], all}$$-->
<!--Robot Land, no human edits to "END"-->
{{InOut|
n = 8 |
in = <nowiki>Length /@ {Union[Jones[#][q]& /@ all], all}</nowiki> |
out= <nowiki>{2110, 2226}</nowiki>}}
<!--END-->
<!--END-->


<span id="How is the Jones polynomial computed?">
<span id="How is the Jones polynomial computed?">
==How is the Jones polynomial computed?==
====How is the Jones polynomial computed?====
</span>
</span>


(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 [[ref|Kauffman]]) the definition of the Jones polynomial using the Kauffman bracket <math>\la\cdot\ra</math>:


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 {{ref|Kauffman}}) the definition of the Jones polynomial using the Kauffman bracket <math>\langle\cdot\rangle</math>:
<center><math>

\la\emptyset\ra=1; \qquad
{{Equation|KBDef|<math>
\la\bigcirc L\ra = (-A^2-B^2)\la L\ra; \qquad
\langle\emptyset\rangle=1; \qquad
\la\slashoverback\ra =
\langle\bigcirc L\rangle = (-A^2-B^2)\langle L\rangle; \qquad
A\la\hsmoothing\ra + B\la\smoothing\ra;
\langle\slashoverback\rangle =
</math></center>
A\langle\hsmoothing\rangle + B\langle\smoothing\rangle;
</math>}}


<center><math> J(L) =
<center><math> J(L) =
\left.(-A^3)^{w(L)}\frac{\la L\ra}{\la\bigcirc\ra}\right|_{A\to q^{1/4}},
\left.(-A^3)^{w(L)}\frac{\langle L\rangle}{\langle\bigcirc\rangle}\right|_{A\to q^{1/4}},
</math></center>
</math></center>


here <math>A</math> is a commutative variable, <math>B=A^{-1}</math>, and <math>w(L)</math> is the ''writhe'' of <math>L</math>, the difference <math>n_+-n_-</math> where <math>n_+</math> and <math>n_-</math> count the positive <math>(\overcrossing)</math> and negative <math>(\undercrossing)</math> crossings of <math>L</math> respectively.
here $A$ is a commutative variable, $B=A^{-1}$, and
$w(L)$ is the {\em writhe} of $L$, the difference $n_+-n_-$ where
$n_+$ and $n_-$ count the positive ($\overcrossing$) and negative
($\undercrossing$) crossings of $L$ respectively.


<center>[[Image:PDForTrefoil.gif|none|frame|<tt><nowiki>PD[X[1,4,2,5], X[3,6,4,1], X[5,2,6,3]]</nowiki></tt> and <tt>P[1,4] P[1,5] P[2,4] P[2,6] P[3,5] P[3,6]</tt>]]</center>
\index{trefoil}
Just for concreteness, let us start by fixing $L$ to be the trefoil
knot (see Figure~\ref{fig:PDTrefoil}):


Just for concreteness, let us start by fixing <math>L</math> to be the trefoil knot shown above:
\begin{figure}
\htmlimage{}
\centering{
\includegraphics[height=3cm]{figs/PDTrefoil.eps}
}
\caption{
{\tt PD[X[1,4,2,5], X[3,6,4,1], X[5,2,6,3]]} and
{\tt P[1,4] P[1,5] P[2,4] P[2,6] P[3,5] P[3,6]}.
} \label{fig:PDTrefoil}
\end{figure}


<!--$$L = PD[Knot[3, 1]]$$-->
<!--$$L = PD[Knot[3, 1]]$$-->
<!--Robot Land, no human edits to "END"-->
{{InOut|
n = 9 |
in = <nowiki>L = PD[Knot[3, 1]]</nowiki> |
out= <nowiki>PD[X[1, 4, 2, 5], X[3, 6, 4, 1], X[5, 2, 6, 3]]</nowiki>}}
<!--END-->
<!--END-->


Our first task is to perform the replacement $\la\slashoverback\ra
Our first task is to perform the replacement <math>\langle\slashoverback\rangle\to A\langle\hsmoothing\rangle + B\langle\smoothing\rangle</math> on all crossings of
<math>L</math>. By our conventions (see [[Planar Diagrams]]) the edges
\to A\la\hsmoothing\ra + B\la\smoothing\ra$ on all crossings of
around a crossing <math>X_{abcd}</math> are labeled <math>a</math>, <math>b</math>, <math>c</math> and <math>d</math>: <math>{}^c_d\slashoverback{}_a^b</math>. Labeling the smoothings <math>(\hsmoothing, \ \smoothing)</math> in the same way, <math>{}^c_d\hsmoothing{}_a^b</math> and <math>{}^c_d\smoothing{}_a^b</math>, we are lead to the symbolic replacement rule <math>X_{abcd}\to AP_{ad}P_{bc}+BP_{ab}P_{cd}</math>. Let us apply this rule to <math>L</math>, switch to a multiplicative notation and expand:
$L$. By our conventions (see Section~\ref{subsec:PD}) the edges
around a crossing $X_{abcd}$ are labeled $a$, $b$, $c$ and $d$:
$\v^c_d\text{\large$\slashoverback$}\v_a^b$. Labeling $\hsmoothing$ and
$\smoothing$ in the same way, $\v^c_d\text{\large$\hsmoothing$}\v_a^b$
and $\v^c_d\text{\large$\smoothing$}\v_a^b$, we are lead to the symbolic
replacement rule $X_{abcd}\to AP_{ad}P_{bc}+BP_{ab}P_{cd}$. Let us apply
this rule to $L$, switch to a multiplicative notation and expand:


<!--$$t1 = L /. X[a_,b_,c_,d_] :> A P[a,d] P[b,c] + B P[a,b] P[c,d]$$-->
<!--$$t1 = L /. X[a_,b_,c_,d_] :> A P[a,d] P[b,c] + B P[a,b] P[c,d]$$-->
<!--Robot Land, no human edits to "END"-->
{{InOut|
n = 10 |
in = <nowiki>t1 = L /. X[a_,b_,c_,d_] :> A P[a,d] P[b,c] + B P[a,b] P[c,d]</nowiki> |
out= <nowiki>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]]</nowiki>}}
<!--END-->
<!--END-->


<!--$$t2 = Expand[Times @@ t1]$$-->
<!--$$t2 = Expand[Times @@ t1]$$-->
<!--Robot Land, no human edits to "END"-->
{{InOut|
n = 11 |
in = <nowiki>t2 = Expand[Times @@ t1]</nowiki> |
out= <nowiki> 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]</nowiki>}}
<!--END-->
<!--END-->


In the above expression the product <tt>P[1,4] P[1,5] P[2,4] P[2,6] P[3,5] P[3,6]</tt> represents a path in which <tt>1</tt> is connected to <tt>4</tt>, <tt>1</tt> is connected to <tt>5</tt>, <tt>2</tt> is connected to <tt>4</tt>, etc. (see the right half of the figure above). We simplify such paths by repeatedly applying the rules <math>P_{ab}P_{bc}\to P_{ac}</math> and <math>P^2_{ab}\to P_{aa}</math>:
In the above expression the product {\tt P[1,4] P[1,5] P[2,4] P[2,6]
P[3,5] P[3,6]} represents a path in which {\tt 1} is connected to {\tt
4}, {\tt 1} is connected to {\tt 5}, {\tt 2} is connected to {\tt 4},
etc. (see Figure~\ref{fig:PDTrefoil}). We simplify such paths by
repeatedly applying the rules $P_{ab}P_{bc}\to P_{ac}$ and $P^2_{ab}\to
P_{aa}$:


<!--$$t3 = t2 //. {P[a_,b_]P[b_,c_] :> P[a,c], P[a_,b_]^2 :> P[a,a]}$$-->
<!--$$t3 = t2 //. {P[a_,b_]P[b_,c_] :> P[a,c], P[a_,b_]^2 :> P[a,a]}$$-->
<!--Robot Land, no human edits to "END"-->
{{InOut|
n = 12 |
in = <nowiki>t3 = t2 //. {P[a_,b_]P[b_,c_] :> P[a,c], P[a_,b_]^2 :> P[a,a]}</nowiki> |
out= <nowiki> 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]</nowiki>}}
<!--END-->
<!--END-->


To complete the computation of the Kauffman bracket, all that remains is to
To complete the computation of the Kauffman bracket, all that remains is to replace closed cycles (paths of the form <math>P_{aa}</math> by <math>-A^2-B^2</math>, to replace <math>B</math> by <math>A^{-1}</math>, and to simplify:
replace closed cycles (paths of the form $P_{aa}$ by $-A^2-B^2$, to replace
$B$ by $A^{-1}$, and to simplify:


<!--$$t4 = Expand[t3 /. P[a_,a_] -> -A^2-B^2 /. B -> 1/A]$$-->
<!--$$t4 = Expand[t3 /. P[a_,a_] -> -A^2-B^2 /. B -> 1/A]$$-->
<!--Robot Land, no human edits to "END"-->
{{InOut|
n = 13 |
in = <nowiki>t4 = Expand[t3 /. P[a_,a_] -> -A^2-B^2 /. B -> 1/A]</nowiki> |
out= <nowiki> -9 1 3 7
-A + - + A + A
A</nowiki>}}
<!--END-->
<!--END-->


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


<!--$$KB0[pd_] := Expand[\n
<!--$$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]]\n //. {P[a_,b_]P[b_,c_] :> P[a,c], P[a_,b_]^2 :> P[a,a], P[a_,a_] -> -A^2-1/A^2}\n
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^2-1/A^2}]$$-->
]$$-->
<!--Robot Land, no human edits to "END"-->
{{In|
n = 14 |
in = <nowiki>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^2-1/A^2}]</nowiki>}}
<!--END-->
<!--END-->


<!--$$t4 = KB0[PD[Knot[3, 1]]]$$-->
<!--$$t4 = KB0[PD[Knot[3, 1]]]$$-->
<!--Robot Land, no human edits to "END"-->
{{InOut|
n = 15 |
in = <nowiki>t4 = KB0[PD[Knot[3, 1]]]</nowiki> |
out= <nowiki> -9 1 3 7
-A + - + A + A
A</nowiki>}}
<!--END-->
<!--END-->


We will skip the uninteresting code for the computation of the writhe here;
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 <math>L</math> the result is <math>-3</math>, and hence the Jones polynomial of <math>L</math> is given by
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 $L$ the
result is $-3$, and hence the Jones polynomial of $L$ is given by


<!--$$(-A^3)^(-3) * t4 / (-A^2-1/A^2) /. A -> q^(1/4) // Simplify // Expand$$-->
<!--$$(-A^3)^(-3) * t4 / (-A^2-1/A^2) /. A -> q^(1/4) // Simplify // Expand$$-->
<!--Robot Land, no human edits to "END"-->
{{InOut|
n = 16 |
in = <nowiki>(-A^3)^(-3) * t4 / (-A^2-1/A^2) /. A -> q^(1/4) // Simplify // Expand</nowiki> |
out= <nowiki> -4 -3 1
-q + q + -
q</nowiki>}}
<!--END-->
<!--END-->


{{Knot Image|L11a548|gif}}
\begin{figure}
\htmlimage{}
\centering{
\includegraphics[height=3cm]{figs/L11a548.eps}
}
\caption{
The link \hlink{../Links/11a548.html}{L11a548}.
}
\end{figure}


At merely 3 lines of code, our program is surely nice and elegant. But it is very slow:
%<* InOut@"Timing[KB0[PD[Link[11, Alternating, 548]]]]"; *>


<!--$$time0 = Timing[KB0[PD[Link[11, Alternating, 548]]]]$$-->
At merely 3 lines of code, our program is surely nice and elegant. But at
<!--Robot Land, no human edits to "END"-->
<* time1=Out[0][[1,1]] *> seconds for an 11 crossing link, it is very slow:
{{InOut|

n = 17 |
<* InOut[0] *>
in = <nowiki>time0 = Timing[KB0[PD[Link[11, Alternating, 548]]]]</nowiki> |
\vskip 6pt
out= <nowiki> -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</nowiki>}}
<!--END-->


Here's the much faster alternative employed by {\tt KnotTheory`}:
Here's the much faster alternative employed by <code>KnotTheory`</code>:


<!--$$KB1[pd_PD] := KB1[pd, {}, 1];
<!--$$
KB1[pd_PD] := KB1[pd, {}, 1];\n
KB1[pd_PD, inside_, web_] := Module[
{pos = First[Ordering[Length[Complement[List @@ #, inside]]& /@ pd]]},
KB1[pd_PD, inside_, web_] := Module[\n
pd[[pos]] /. X[a_,b_,c_,d_] :> KB1[
{pos = First[Ordering[Length[Complement[List @@ #, inside]]& /@ pd]]},\n
Delete[pd, pos],
pd[[pos]] /. X[a_,b_,c_,d_] :> KB1[\n
Delete[pd, pos],\n
Union[inside, {a,b,c,d}],
Union[inside, {a,b,c,d}],\n
Expand[web*(A P[a,d] P[b,c]+1/A P[a,b] P[c,d])] //. {
Expand[web*(A P[a,d] P[b,c]+1/A P[a,b] P[c,d])] //. {\n
P[e_,f_]P[f_,g_] :> P[e,g], P[e_,_]^2 :> P[e,e], P[e_,e_] -> -A^2-1/A^2
}
P[e_,f_]P[f_,g_] :> P[e,g], P[e_,_]^2 :> P[e,e], P[e_,e_] -> -A^2-1/A^2\n
}\n
]
];
]\n
KB1[PD[],_,web_] := Expand[web]$$-->
];\n
<!--Robot Land, no human edits to "END"-->
KB1[PD[],_,web_] := Expand[web]"$$-->
{{In|
n = 18 |
in = <nowiki>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^2-1/A^2
}
]
];
KB1[PD[],_,web_] := Expand[web]</nowiki>}}
<!--END-->
<!--END-->


<!--$$Timing[KB1[PD[Link[11, Alternating, 548]]]]$$-->
<!--$$time1 = Timing[KB1[PD[Link[11, Alternating, 548]]]]$$-->
<!--Robot Land, no human edits to "END"-->
{{InOut|
n = 19 |
in = <nowiki>time1 = Timing[KB1[PD[Link[11, Alternating, 548]]]]</nowiki> |
out= <nowiki> -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 }</nowiki>}}
<!--END-->
<!--END-->


(So on [[L11a548]] <code>KB1</code> is <!--$time0[[1,1]]$--><!--Robot Land, no human edits to "END"--> -23 5 10 -3 5 13 17 21 25
(So on the link \hlink{../Links/11a548.html}{L11a548} \verb"KB1" is
{1., A + --- + -- + A + 6 A + 6 A + 5 A - 5 A + 4 A - A }[[1,1]]
$<*time1*>/<*time2=Out[0][[1,1]]*>\sim<*Round[time1/time2]*>$ times
15 7
faster than \verb"KB0".)
A A<!--END-->/<!--$time1[[1,1]]$--><!--Robot Land, no human edits to "END"--> -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<!--END--> ~ <!--$Round[time0[[1,1]]/time1[[1,1]]]$--><!--Robot Land, no human edits to "END"--> -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<!--END--> times faster than <code>KB0</code>.)


The idea here is to maintain a ``computation front'', a planar domain
The idea here is to maintain a "computation front", a planar domain
which starts empty and gradualy increases until the whole link diagram is
which starts empty and gradualy increases until the whole link diagram is
enclosed. Within the front, the rules defining the Kauffman bracket,
enclosed. Within the front, the rules defining the Kauffman bracket,
Equation~\eqref{eq:KBDef}, are applied and the result is expanded as much
Equation {{Equation Ref|KBDef}}, are applied and the result is expanded as much
as possible. Outside of the front the link diagram remains untouched. At
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
every step we choose a crossing outside the front with the most legs
inside and ``conquer'' it --- apply the rules of~\eqref{eq:KBDef} and
inside and "conquer" it -- apply the rules of {{Equation Ref|KBDef}} and
expand again. As our new outpost is maximally connected to our old
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
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
hence the size of the "web" within our front remains as small as
possible and thus quick to manipulate.
possible and thus quick to manipulate.


In further detail, the routine \verb"KB1[pd, inside, web]" computes the
In further detail, the routine <code>KB1[pd, inside, web]</code> computes the
Kauffman bracket assuming the labels of the edges inside the front are in
Kauffman bracket assuming the labels of the edges inside the front are in
the variable {\tt inside}, the already-computed inside of the front is in
the variable <code>inside</code>, the already-computed inside of the front is in
the variable {\tt web} and the part of the link diagram yet untouched is
the variable <code>web</code> and the part of the link diagram yet untouched is
{\tt pd}. The single argument \verb"KB1[pd]" simply calls
<code>pd</code>. The single argument <code>KB1[pd]</code> simply calls
\verb"KB1[pd, inside, web]" with an empty {\tt inside} and with {\tt web}
<code>KB1[pd, inside, web]</code> with an empty <code>inside</code> and with <code>web</code> set to 1. The three argument <code>KB1[pd, inside, web]</code> finds the position of the crossing maximmally connected to the front using the somewhat
set to 1. The three argument \verb"KB1[pd, inside, web]" finds the position
of the crossing maximmally connected to the front using the somewhat
cryptic assignment
cryptic assignment


pos = First[Ordering[Length[Complement[List @@ #, inside]]& /@ pd]]}
\begin{verbatim}

pos = First[Ordering[Length[Complement[List @@ #, inside]]& /@ pd]]}
<code>KB1[pd, inside, web]</code> then recursively calls
\end{verbatim}
itself with that crossing removed from <code>pd}, with its legs
added to the <code>inside</code>, and with <code>web</code> updated in accordance
with {{Equation Ref|KBDef}}. Finally, when <code>pd</code> is empty, the output is
simply the value of <code>web</code>.


{{note|Kauffman}} L. H. Kauffman, ''On knots'', Princeton Univ. Press, Princeton, 1987.
\noindent \verb"KB1[pd, inside, web]" then recursively calls
itself with that crossing removed from {\tt pd}, with its legs
added to the {\tt inside}, and with {\tt web} updated in accordance
with~\eqref{eq:KBDef}. Finally, when {\tt pd} is empty, the output is
simply the value of {\tt web}.

Latest revision as of 19:53, 8 August 2013


KnotTheory`/Navigation 

The Mathematica Package KnotTheory`

(For In[1] see Setup)

In[2]:= ?Jones
Jones[L][q] computes the Jones polynomial of a knot or link L as a function of the variable q.

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.gif
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]
Failed to parse (unknown function "\slashoverback"): {\displaystyle \langle\emptyset\rangle=1; \qquad \langle\bigcirc L\rangle = (-A^2-B^2)\langle L\rangle; \qquad \langle\slashoverback\rangle = A\langle\hsmoothing\rangle + B\langle\smoothing\rangle; }

here is a commutative variable, , and is the writhe of , the difference where and count the positive Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\overcrossing)} and negative Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\undercrossing)} crossings of respectively.

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

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^2-B^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^2-1/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^2-1/A^2) /. A -> q^(1/4) // Simplify // Expand
Out[16]= -4 -3 1 -q + q + - q
L11a548.gif
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^2-1/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 already-computed 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.

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