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The Alexander-Conway Polynomial

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(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.
8 18.gif
8_18

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]= 1
K11a99.gif
K11a99
K11a277.gif
K11a277

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}