A polynomial invariant for knots via von Neumann algebras. (English) Zbl 0564.57006

The author introduces a new polynomial invariant \(V_ L(t)\) for tame oriented links via certain representations of the braid group. That the invariant depends only on the closed braid is a direct consequence of Markov’s theorem and a certain trace formula, which was discovered because of the uniqueness of the trace on certain von Neumann algebras. There is an alternate way to calculate \(V_ L(t)\) without converting L into a closed braid, using only a Conway type relation; \(V_{unknot}=1\) and 1/t \(V_{L-}-t V_{L+}=(\sqrt{t}-1/\sqrt{t})V_ L\). This is also interesting from a view point of formal knot theory. The author gives many results using this invariant. For an example, \(V_{L\sim}(t)=V_ L(1/t)\) where \(L\sim\) means the mirror image of L, \(V_{L_ 1\#L_ 2}=V_{L_ 1}\cdot V_{L_ 2}\) where # means a connected sum of links, \(V_ L(-1)=\Delta_ L(-1)\) where \(\Delta_ L\) means the Alexander polynomial, \(V_ L(1)=(-2)^{p-1}\) where p is the number of components of L, \(V_ K(e^{2\pi i/3})=1\) and d/dt \(V_ K(1)=0\) if K is a knot. If K is a knot and \(| \Delta_ K(i)| >3\), then k cannot be represented as a closed 3 braid. If K is a knot and \(\Delta (e^{2\pi i/5})>6.5\), then K cannot be represented as a closed 4 braid.
Reviewer: Y.Nakanishi


57M25 Knots and links in the \(3\)-sphere (MSC2010)
46L99 Selfadjoint operator algebras (\(C^*\)-algebras, von Neumann (\(W^*\)-) algebras, etc.)
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