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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

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