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.

中文翻译：

---

标记的奇异结

我们引入了标记的奇异结和等效标记的 4 价刚性顶点空间图。标记奇异结是具有标记奇异点的奇异结。这些结被认为受同位素保留标签的影响。我们提供了一个类似于 Henrich 和 Kauffman 在 [A. Henrich 和 LH Kauffman，伪结、奇异结和刚性顶点空间图的缠结插入不变量，当代。数学。689 (2017) 1-10] 通过在标记的奇异点处插入有理缠结，将通常的结不变量扩展到我们的奇异结类。我们表明我们可以使用标记奇异结的不变量来服务于通常的奇异结。还介绍和讨论了带标签的框架假结。