On a Map from Pure Braids to Knots

maps are compatible with the inclusions P2n+1 →֒ P2n+3 so they extend to a map S : P∞ → K. Here by P∞ we understand the inductive limit of the sequence of inclusions Pi →֒ Pi+1. The construction and, as we will see later, some properties of the map S resemble those of the plat closure which sends braids with even number of strands to links. (For the definition and properties of the plat closure see [B1, B2].) Indeed, if tn denotes the 2n-strand braid pictured on Figure 2, then for any x ∈ P2n+1 the (unoriented) knot S(x) is equivalent to the knot, obtained by taking the image of x in P2n+2 under the standard inclusion, multiplying by tn+1 on the left (i.e. on the top) and taking the plat closure. However, if we are interested in knots rather than links the map S is more convenient than the plat closure. The most obvious difference is the behaviour under stabilization maps and tensor products. Adding two unbraided strands to a braid changes its image under the plat closure by adding an unknotted and unlinked component, while the image of the short-circuit map does not change. As for tensor (external) products, the plat closure sends a product of braids to the distant union