Topology in Molecular Biology

88 L.H. Kauffman and S. Lambropoulou

S

double

resmoothing

S `

Fig. 5.14.A double resmoothing

[0] - [ ] interchange

S S `

contribute to N contribute to D

A

A

-1

Fig. 5.15.Non-trivial states

for some pair of statesS, S′,then it follows from the first claim that it is
true for all pairs of states, and that〈N(T)〉=ωp, 〈D(T)〉=ω′q, p, q∈Z
andω/ω′=〈S〉/〈S′〉=±i.Hence〈N(T)〉/〈D(T)〉=±ip/q,wherep/qis a
rational number (orq= 0). This will complete the proof thatF(T)isrealor
∞.
To see this second claim we consider specific pairs of states as in Fig. 5.15.
We have illustrated representative statesSandS′of the tangleT. We obtain
S′fromSby resmoothing at one site that changesSfrom an [∞] tangle to
the [0] tangle underlyingS′. Then〈S〉/〈S′〉=A±^2 =±i.If there is no such
resmoothing site available, then it follows thatD(T) is a disjoint union of
two diagrams, and hence〈D(T)〉=0andF(T)=∞.This does complete the
proof of Statement 1.