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Let  X  be an aspherical polyhedron of the homotopy type of the figure-eight and let  f  :  X  →  X  be a self-map. The Wagner algorithm  [Trans. Amer. Math. Soc. 351 (1999), 41–62] provides computations for the Nielsen number of self-maps of  X  satisfying the remnant condition. If  f  is without remnant, then using the concept of mutant by Jiang [Math. Ann. 311 (1998), 467–479] we may assume that  f  # ( b ) is an initial segment of  f  # ( a ), where  f  #  is the induced endomorphism of π 1 ( X ) and  a ,  b  are generators of π 1 ( X ). Let  f  # ( b ) =  U  and  f  # ( a ) =  U n   R , where  n  is the maximal such positive integer. If  R  is not an initial segment of  U , we say that  f  is of Type  Y . In this paper, we prove that if  f  is of Type  Y , then  f  can be mutated either to a map that has remnant or to an exceptional form for which we can calculate the Nielsen number directly. Not all self-maps of  X  are of Type  Y . However, making use of the results in this paper, an algorithm is presented by Kim [J. Pure Appl. Algebra 216 (2012), 1652–1666] that does compute the Nielsen number for all self-maps of  X .

Nielsen numbers of maps of aspherical figure-eight type polyhedra