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Knot projections with reductivity two

Reductivity of knot projections refers to the minimum number of splices of double points needed to obtain reducible knot projections. Considering the type and method of splicing (Seifert type splice or non-Seifert type splice, recursively or simultaneously), we can obtain four reductivities containing Shimizu's reductivity, three of which are new. In this paper, we determine knot projections with reductivity two for all four of the definitions. We also provide easily calculated lower bounds for some reductivities. Further, we detail properties of each reductivity, and describe relationships among the four reductivities with examples.