### Quick description

If you find a surgery formula for an invariant, and if you know its value for the unknot (or for the unlink), and if you suspect that it coincides with an invariant given in terms of a canonical system of curves on a Seifert surface, then the following trick may be useful. Choose a band projection for the Seifert surface, consisting of a disc with bands attached in `commutator' order (add diagram). Then clasper calculus gives you a surgery presentation of the knot in the complement of a standard unknot, and the linking matrix for the leaves of the claspers which do not intersect a disc bounded by the unknot coincides with the corresponding Seifert matrix for the knot (or link). This presentation can be further simplified (cite). Plug your surgery formula in, and get an invariant given in terms of canonical system of curves of the Seifert surface (for example, it might be given in terms of the Seifert matrix).

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