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We work out the details of a correspondence observed by Goodwillie betweencosimplicial spaces and good functors from a category of open subsets of theinterval to the category of compactly generated weak Hausdorff spaces. Usingthis, we compute the first page of the integral Bousfield--Kan homotopyspectral sequence of the tower of fibrations given by the Taylor tower of theembedding functor associated to the space of long knots. Based on the methodsin [Con08], we give a combinatorial interpretation of the differentials $d^1$mapping into the diagonal terms, by introducing the notion of $(i, n)$-markedunitrivalent graphs.

Goodwillie's cosimplicial model for the space of long knots and its applications