The dihedral genus of a knot

Cahn, Patricia¹ ; Kjuchukova, Alexandra²

¹Department of Mathematics and Statistics, Smith College, Northampton, MA, United States
²Max Planck Institute for Mathematics, Bonn, Germany

Algebraic and Geometric Topology, Tome 20 (2020) no. 4, pp. 1939-1963

Résumé

Let K ⊂ S3 be a Fox p–colored knot and assume K bounds a locally flat surface S ⊂ B4 over which the given p–coloring extends. This coloring of S induces a dihedral branched cover X → S4. Its branching set is a closed surface embedded in S4 locally flatly away from one singularity whose link is K. When S is homotopy ribbon and X a definite four-manifold, a condition relating the signature of X and the Murasugi signature of K guarantees that S in fact realizes the four-genus of K. We exhibit an infinite family of knots Km with this property, each with a Fox 3–colored surface of minimal genus m. As a consequence, we classify the signatures of manifolds X which arise as dihedral covers of S4 in the above sense.

Détail

DOI : 10.2140/agt.2020.20.1939

Classification : 57M12, 57M25, 57Q60
Keywords: knot, branched cover, ribbon genus, trisection

Bibliographie

[1] R Blair, P Cahn, A Kjuchukova, Blowing up dihedral covers, in preparation

[2] P Cahn, A Kjuchukova, Linking numbers in three-manifolds, preprint (2016)

[3] P Cahn, A Kjuchukova, Singular branched covers of four-manifolds, preprint (2017)

[4] P Cahn, A Kjuchukova, Computing ribbon obstructions for colored knots, preprint (2018)

[5] S E Cappell, J L Shaneson, Linking numbers in branched covers, from: "Four-manifold theory" (editors C Gordon, R Kirby), Contemp. Math. 35, Amer. Math. Soc. (1984) 165 | DOI

[6] A J Casson, C M Gordon, Cobordism of classical knots, from: "À la recherche de la topologie perdue" (editors L Guillou, A Marin), Progr. Math. 62, Birkhäuser, Boston (1986) 181

[7] R H Fox, Some problems in knot theory, from: "Topology of –manifolds and related topics" (editor J M. K. Fort), Prentice-Hall, Englewood Cliffs, NJ (1962) 168

[8] D Gay, R Kirby, Trisecting 4–manifolds, Geom. Topol. 20 (2016) 3097 | DOI

[9] C Geske, A Kjuchukova, J L Shaneson, Signatures of topological branched covers, preprint (2019)

[10] C M Gordon, R A Litherland, On the signature of a link, Invent. Math. 47 (1978) 53 | DOI

[11] H M Hilden, Every closed orientable 3–manifold is a 3–fold branched covering space of S3, Bull. Amer. Math. Soc. 80 (1974) 1243 | DOI

[12] A Kjuchukova, On the classification of irregular dihedral branched covers of four-manifolds, PhD thesis, University of Pennsylvania (2015)

[13] A Kjuchukova, Dihedral branched covers of four-manifolds, Adv. Math. 332 (2018) 1 | DOI

[14] A Kjuchukova, K Orr, Knots arising as singularities on branched covers between four-manifolds, in preparation

[15] J Meier, A Zupan, Bridge trisections of knotted surfaces in S4, Trans. Amer. Math. Soc. 369 (2017) 7343 | DOI

[16] K Murasugi, On a certain numerical invariant of link types, Trans. Amer. Math. Soc. 117 (1965) 387 | DOI