# Nomenclature for links of 2-bouquet homotopic class

I've already understood the basic nomenclature for knot and links, especially links consisting trivial knots (circle). However, things got much complicated when I was working on the case when circles are replaced by 2-bouquet graphs (which may be regarded homotopic to two circle glued together). For example, how do you name these links below by link diagrams or projections? enter image description here

And further, how could I find out their linking numbers?

• I am wondering that whether Seifert's algorithm or matrix is still applicable here? Commented Mar 22, 2023 at 2:49
• These are sometimes called welded links or knots, first defined here but I am sure there are other more recent sources you might be interested in now. Commented Mar 22, 2023 at 13:45

In fact, you can use some kind of "homotopy trick"here. As this example is clearly easy to avoid further confusion, there is an approach to compute linking numbers for simple graphs without complicated intra- or interwoven feature: (linking number of graphs) The linking number λ(G) of a graph G is: $$λ(G) = ∑_{i=j} λ(G_i, G_j)$$,
where $$λ(G_i,G_j) = ∑|λ(z_p,z_q)|$$ for ball basis cycles $$z_p,z_q$$ in different components $$G_i,G_j$$,respectively.