## 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

### Citations:

Zbl 0483.57004; Zbl 1132.57004; Zbl 1188.57005; Zbl 1303.57006
Full Text:

### References:

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