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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?

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    $\begingroup$ I am wondering that whether Seifert's algorithm or matrix is still applicable here? $\endgroup$
    – Gokouu
    Commented Mar 22, 2023 at 2:49
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    $\begingroup$ 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. $\endgroup$
    – N. Owad
    Commented Mar 22, 2023 at 13:45

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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.

enter image description here

The linking number is computed only between pairs of components following Seifert’s original definition. Linked cycles within the same component may be unlinked by a homotopy (Prasolov, 1995).

REF: 1 Prasolov, V. V., & Prasolov, V. V. E. (1995). Intuitive topology (No. 4). American Mathematical Soc..

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