# The Determinant and the Signature

(For In[1] see Setup)

 In[2]:= ?KnotDet KnotDet[K] returns the determinant of a knot K.
 In[3]:= ?KnotSignature KnotSignature[K] returns the signature of a knot K.

Thus, for example, the knots 5_1 and 10_132 have the same determinant (and even the same Alexander and Jones polynomials), but different signatures:

 In[4]:= KnotDet /@ {Knot[5, 1], Knot[10, 132]} Out[4]= {5, 5}
 In[5]:= { Equal @@ (Jones[#][q]& /@ {Knot[5, 1], Knot[10, 132]}), Equal @@ (Alexander[#][t]& /@ {Knot[5, 1], Knot[10, 132]}) } Out[5]= {True, True}
 In[6]:= KnotSignature /@ {Knot[5, 1], Knot[10, 132]} Out[6]= {-4, 0}

In August 2005 somebody emailed Dror a question about knot colouring, which amounted to "find the first knot (other than the unknot) whose determinant is ${\displaystyle \pm 1}$". So on September 2nd Dror typed

 In[7]:= Select[AllKnots[], Abs[KnotDet[#]] == 1 &] Out[7]= {Knot[0, 1], Knot[10, 124], Knot[10, 153], Knot[11, NonAlternating, 34], Knot[11, NonAlternating, 42], Knot[11, NonAlternating, 49], Knot[11, NonAlternating, 116]}

Hence the first few knots that are not ${\displaystyle k}$-colourable for any ${\displaystyle k}$ are 10_124, 10_153, K11n34, K11n42, K11n49 and K11n116.