The Jones Polynomial

KnotTheory`/Navigation  The Mathematica Package KnotTheory` Acknowledgement Setup Naming and Enumeration Presentations Planar Diagrams Some further details Gauss Codes DT (Dowker-Thistlethwaite) Codes Braid Representatives MorseLink Presentations Arc Presentations Conway Notation Graphical Input Graphical Output Drawing Planar Diagrams How does it work? Drawing MorseLink Presentations Drawing Braids Structure and Operations Invariants Invariants from Braid Theory Three Dimensional Invariants The Alexander-Conway Polynomial The Multivariable Alexander Polynomial The Determinant and the Signature The Jones Polynomial How is the Jones polynomial computed? The Coloured Jones Polynomials The A2 Invariant Quantum knot invariants The HOMFLY-PT Polynomial The Kauffman Polynomial Finite Type (Vassiliev) Invariants Khovanov Homology Heegaard Floer Knot Homology R-Matrix Invariants Extras Included with KnotTheory` Drawing with TubePlot Standalone_TubePlot Using the LinKnot package WikiLink QuantumGroups` Lightly Documented Features A Sample KnotTheory` Session Further Usage Example Prime Links with a Non-Prime Component "Rubberband" Brunnian Links Identifying Knots within a List Cabling Burau's Theorem Further Knot Theory Software KnotPlot Knotscape Xknots Extending/Modifying KnotTheory` Bugs How to Edit this Manual...

(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

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

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