Nakamura, Inasa Simplifying branched covering surface-knots by chart moves involving black vertices. (English) Zbl 1402.57022 Ill. J. Math. 61, No. 3-4, 497-515 (2017). In the author’s previous paper [J. Knot Theory Ramifications 27, No. 5, Article ID 1850031, 29 p. (2018; Zbl 1394.57021)], it was shown that one can deform a branched covering surface-knot (formerly called a 2-dimensional braid) over an oriented surface-knot \(F\) (unknotted in the standard form) to a simplified form. The present paper investigates further such simplifications of branched covering surface-knots when they have branch points. This is done by addition of finitely many 1-handles of given form to appropriate places in \(F\). Each of these 1-handles is called a 1-handle with chart loop or simply a 1-handle. The obtained form is called the simplified form and the necessary minimal number of 1-handles, denoted \(u_w\), is called the simplified number. It is proved that for a branched covering surface-knot of degree \(N\), with non-zero black vertices and zero white vertices, it holds that \(u_w \leq N-2.\) Reviewer: Ivan Ivanšić (Zagreb) Cited in 1 ReviewCited in 2 Documents MSC: 57Q45 Knots and links in high dimensions (PL-topology) (MSC2010) 57M25 Knots and links in the \(3\)-sphere (MSC2010) 57Q35 Embeddings and immersions in PL-topology Keywords:surface knot; 2-dimensional braid; branched covering; chart moves; branch point; black vertices; 1-handle Citations:Zbl 1394.57021 × Cite Format Result Cite Review PDF Full Text: arXiv Euclid References: [1] J. Boyle, The turned torus knot in \(S^4\), J. Knot Theory Ramifications 2 (1993), 239–249. · Zbl 0995.57502 · doi:10.1142/S0218216593000155 [2] J. S. Carter, S. Kamada and M. Saito, Surfaces in 4-space, Low-dimensional topology III, Encyclopaedia of Mathematical Sciences, vol. 142, Springer-Verlag, Berlin, 2004. · Zbl 1078.57001 · doi:10.1007/978-3-662-10162-9 [3] S. Kamada, Surfaces in \(R^4\) of braid index three are ribbon, J. Knot Theory Ramifications 1 (1992), 137–160. · Zbl 0763.57013 · doi:10.1142/S0218216592000082 [4] S. Kamada, An observation of surface braids via chart description, J. Knot Theory Ramifications 4 (1996), 517–529. · Zbl 0889.57011 · doi:10.1142/S0218216596000308 [5] S. Kamada, Braid and knot theory in dimension four, Math. Surveys and Monographs, vol. 95, Amer. Math. Soc., 2002. · Zbl 0993.57012 · doi:10.1090/surv/095 [6] S. Kamada, Surface-knots in 4-space, Springer Monographs in Mathematics, Springer, 2017. · Zbl 1362.57001 · doi:10.1007/978-981-10-4091-7 [7] C. Livingston, Stably irreducible surfaces in \(S^4\), Pacific J. Math. 116 (1983), 77–84. · Zbl 0559.57018 · doi:10.2140/pjm.1985.116.77 [8] I. Nakamura, Surface links which are coverings over the standard torus, Algebr. Geom. Topol. 11 (2011), 1497–1540. · Zbl 1230.57022 · doi:10.2140/agt.2011.11.1497 [9] I. Nakamura, Satellites of an oriented surface link and their local moves, Topology Appl. 164 (2014), 113–124. · Zbl 1295.57028 · doi:10.1016/j.topol.2013.12.010 [10] I. Nakamura, On addition of 1-handles with chart loops to 2-dimensional braids, J. Knot Theory Ramifications 26 (2016), \bnumber1650061. · Zbl 1355.57024 · doi:10.1142/S0218216516500619 [11] I. Nakamura, Simplifying branched covering surface-knots by an addition of 1-handles with chart loops, J. Knot Theory Ramifications 27 (2018), \bnumber1850031. · Zbl 1394.57021 · doi:10.1142/S0218216518500311 [12] D. Roseman, Reidemeister-type moves for surfaces in four-dimensional space, Knot theory, Banach Center Publications, vol. 42, Polish Acad. Sci., 1998, pp. 347–380. · Zbl 0906.57010 [13] L. Rudolph, Braided surfaces and Seifert ribbons for closed braids, Comment. Math. Helv. 58 (1983), 1–37. · Zbl 0522.57017 · doi:10.1007/BF02564622 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.