Labeled singular knots

Abstract

We introduce labeled singular knots and equivalently labeled 4-valent rigid vertex spatial graphs. Labeled singular knots are singular knots with labeled singularities. These knots are considered subject to isotopies preserving the labelings. We provide a topological invariant schema similar to that of Henrich and Kauffman in [A. Henrich and L. H. Kauffman, Tangle insertion invariants for pseudoknots, singular knots, and rigid vertex spatial graphs, Contemp. Math. 689 (2017) 1–10] by inserting rational tangles at the labeled singularities to extend usual knot invariants to our class of singular knots. We show that we can use invariants of labeled singular knots to serve usual singular knots. Labeled framed pseudoknots are also introduced and discussed.