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CHARACTERIZATION OF FRAMINGS VIA COHOMOLOGY OF THE KNOT QUANDLE
https://doi.org/10.1142/S0218216511009431Cited by:0 (Source: Crossref)
Abstract
For a knot K with a framing F, we show that the knot rack R(K, F) of (K, F) has the structure of a rack extension of the knot quandle Q(K), via the projection canonically obtained by disregarding its framing. From this fact, we can construct a cohomology class [ϕF] of Q(K) corresponding to the framing: a cohomology class in 
for K non-trivial, or one in 
for K trivial, where wF denotes the writhe of (K, F). Moreover, when K is non-trivial, [ϕF] has a natural decomposition to the sum of 
and 
, where [ψ] is independent to F. This correspondence 
gives a bijection between the set of framings of K and 
.
Keywords:
AMSC: 57M27, 57M25
