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Multivariable signatures, genus bounds, and \(0.5\)-solvable cobordisms. (English) Zbl 1454.57003

Given a a link \(L\) whose components are partitioned into \(\mu\) sublinks – i.e. a \(\mu\)-colored link – one can associate two functions \(\sigma_L, \eta_L \colon (S^1 \setminus \{1\})^{\mu} \to \mathbb{Z}\) called respectively the multivariable signature and nullity functions. The functions \(\sigma_L\) and \(\eta_L\), originally defined by D. Cooper [Lond. Math. Soc. Lect. Note Ser. 48, 51–66 (1982; Zbl 0483.57004)] (see also [D. Cimasoni and V. Florens, Trans. Am. Math. Soc. 360, No. 3, 1223–1264 (2008; Zbl 1132.57004)]) generalize the classical Tristram-Levine signature and nullity functions of an ordinary (i.e. 1-colored) link. Later work of [O. Viro, J. Knot Theory Ramifications 18, No. 6, 729–755 (2009; Zbl 1188.57005)] gave a direct interpretation of these two functions in terms of certain surfaces in the 4-ball bounding the link. In this work, the authors fruitfully apply Viro’s perspective to generalize several results about the Tristram-Levine signature function to the multivariable setting. First, they establish the multivariable signature and nullity functions of the \(\mu\)-colored links \(L\) and \(L'\) can be used to provide bounds on the (negative) Euler characteristic and number of double points of any immersed \(\mu\)-colored surface between \(L\) and \(L'\). This implies that many values of \(\sigma_L\) and \(\mu_L\) are invariant under colored link concordance. The authors go on to show that the multivariable signature and nullity functions are invariant even under \(0.5\)-solvable cobordism, a weaker form of concordance defined by J. C. Cha [Trans. Am. Math. Soc. 366, No. 6, 3241–3273 (2014; Zbl 1303.57006)].

MSC:

57K10 Knot theory
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