7.4 The Determinant and the Signature

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7.4 The Determinant and the Signature

In[1]

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 and $10_{132}$ have the same determinant (and even the same Alexander and Jones polynomials), but different signatures:

**Figure 5:** The knots and $10_{132}$ .
$\begin{figure}\centering { \includegraphics[height=2cm]{figs/5.1.eps} \qquad\includegraphics[height=2cm]{figs/10.132.eps} } \end{figure}$

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}

Dror Bar-Natan 2005-09-14

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}