The Determinant and the Signature

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(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:

5 1.gif
5_1
10 132.gif
10_132
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 \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 k-colourable for any k are 10_124, 10_153, K11n34, K11n42, K11n49 and K11n116.

K11n116.gif
K11n116