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We establish a pseudoisotopy result for embedding spaces in the line of thatof Weiss and Williams for diffeomorphism groups. In other words, for $P\subsetM$ a codimension at least three embedding, we describe the difference in arange of homotopical degrees between the spaces of block and ordinaryembeddings of $P$ into $M$ as a certain infinite loop space involving therelative algebraic $K$-theory of the pair $(M,M-P)$. This range of degrees isthe so-called concordance embedding stable range, which, by recent developmentsof Goodwillie--Krannich--Kupers, is far beyond that of the aforementionedtheorem of Weiss--Williams.  As an application, we give a full description of the homotopy type (away from2 and up to the concordance embedding stable range) of the space of long knotsof codimension at least 3.

A Weiss--Williams theorem for spaces of embeddings and the homotopy type of spaces of long knots