The Alexander-Conway 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]:= ?Alexander Alexander[K][t] computes the Alexander polynomial of a knot K as a function of the variable t. Alexander[K, r][t] computes a basis of the r'th Alexander ideal of K in Z[t].

In[3]:= Alexander::about The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

In[4]:= ?Conway Conway[K][z] computes the Conway polynomial of a knot K as a function of the variable z.

The Alexander polynomial $A(K)$ and the Conway polynomial $C(K)$ of a knot $K$ always satisfy $A(K)(t)=C(K)({\sqrt {t}}-1/{\sqrt {t}})$. Let us verify this relation for the knot 8_18:

In[5]:=	alex = Alexander[Knot[8, 18]][t]
Out[5]=	-3 5 10 2 3 13 - t + -- - -- - 10 t + 5 t - t 2 t t

In[6]:=	Expand[Conway[Knot[8, 18]][Sqrt[t] - 1/Sqrt[t]]]
Out[6]=	-3 5 10 2 3 13 - t + -- - -- - 10 t + 5 t - t 2 t t

The determinant of a knot $K$ is $|A(K)(-1)|$. Hence for 8_18 it is

In[7]:=	Abs[alex /. t -> -1]
Out[7]=	45

Alternatively (see The Determinant and the Signature):

In[8]:=	KnotDet[Knot[8, 18]]
Out[8]=	45

$V_{2}(K)$, the (standardly normalized) type 2 Vassiliev invariant of a knot $K$ is the coefficient of $z^{2}$ in its Conway polynomial:

In[9]:=	Coefficient[Conway[Knot[8, 18]][z], z^2]
Out[9]=	1

Alternatively (see Finite Type (Vassiliev) Invariants),

In[10]:=	Vassiliev[2][Knot[8, 18]]
Out[10]=	0

Sometimes two knots have the same Alexander polynomial but different Alexander ideals. An example is the pair K11a99 and K11a277. They have the same Alexander polynomial, but the second Alexander ideal of the first knot is the whole ring ${\mathbb {Z} }[t]$ while the second Alexander ideal of the second knot is the smaller ideal generated by $3$ and by $1+t$:

In[11]:=

{K1, K2} = {Knot[11, Alternating, 99], Knot[11, Alternating, 277]};

In[12]:=	Alexander[K1] == Alexander[K2]
Out[12]=	True

In[13]:=	Alexander[K1, 2][t]
Out[13]=	{1}

In[14]:=	Alexander[K2, 2][t]
Out[14]=	{3, 1 + t}

Finally, the Alexander polynomial attains 551 values on the 802 knots known to KnotTheory`:

In[15]:=	Length /@ {Union[Alexander[#]& /@ AllKnots[]], AllKnots[]}
Out[15]=	{551, 802}