Widths of surface knots.

*(English)*Zbl 1132.57021Summary: We study surface knots in 4-space by using generic planar projections. These projections have fold points and cusps as their singularities and the image of the singular point set divides the plane into several regions. The width (or the total width) of a surface knot is a numerical invariant related to the number of points in the inverse image of a point in each of the regions. We determine the widths of certain surface knots and characterize those surface knots with small total widths. Relation to the surface braid index is also studied.

##### MSC:

57Q45 | Knots and links in high dimensions (PL-topology) (MSC2010) |

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

##### Keywords:

surface knot; bridge index; width; total width; braid index; spun knot; ribbon surface knot**OpenURL**

##### References:

[1] | P M Akhmet’ev, Smooth immersions of manifolds of small dimension, Mat. Sb. 185 (1994) 3 · Zbl 0842.57026 |

[2] | S Bleiler, M Scharlemann, A projective plane in \(\mathbfR^4\) with three critical points is standard. Strongly invertible knots have property \(P\), Topology 27 (1988) 519 · Zbl 0678.57003 |

[3] | V Carrara, J S Carter, M Saito, Singularities of the projections of surfaces in 4-space, Pacific J. Math. 199 (2001) 21 · Zbl 1051.57033 |

[4] | V L Carrara, M A S Ruas, O Saeki, Maps of manifolds into the plane which lift to standard embeddings in codimension two, Topology Appl. 110 (2001) 265 · Zbl 0971.57036 |

[5] | J S Carter, M Saito, Knotted surfaces and their diagrams, Mathematical Surveys and Monographs 55, American Mathematical Society (1998) · Zbl 0904.57010 |

[6] | T Cochran, Ribbon knots in \(S^4\), J. London Math. Soc. \((2)\) 28 (1983) 563 · Zbl 0501.57009 |

[7] | T Fukuda, Topology of folds, cusps and Morin singularities, Academic Press (1988) 331 · Zbl 0658.58012 |

[8] | D Gabai, Foliations and the topology of \(3\)-manifolds. III, J. Differential Geom. 26 (1987) 479 · Zbl 0639.57008 |

[9] | M Golubitsky, V Guillemin, Stable mappings and their singularities, Graduate Texts in Mathematics 14, Springer (1973) · Zbl 0294.58004 |

[10] | F Hosokawa, A Kawauchi, Proposals for unknotted surfaces in four-spaces, Osaka J. Math. 16 (1979) 233 · Zbl 0404.57020 |

[11] | S Kamada, Surfaces in \(\mathbfR^4\) of braid index three are ribbon, J. Knot Theory Ramifications 1 (1992) 137 · Zbl 0763.57013 |

[12] | S Kamada, Braid and knot theory in dimension four, Mathematical Surveys and Monographs 95, American Mathematical Society (2002) · Zbl 0993.57012 |

[13] | A Kawauchi, On pseudo-ribbon surface-links, J. Knot Theory Ramifications 11 (2002) 1043 · Zbl 1029.57026 |

[14] | H I Levine, Mappings of manifolds into the plane, Amer. J. Math. 88 (1966) 357 · Zbl 0146.45101 |

[15] | J N Mather, Generic projections, Ann. of Math. \((2)\) 98 (1973) 226 · Zbl 0242.58001 |

[16] | O Saeki, Y Takeda, Canceling branch points and cusps on projections of knotted surfaces in 4-space, Proc. Amer. Math. Soc. 132 (2004) 3097 · Zbl 1052.57033 |

[17] | M Scharlemann, Smooth spheres in \(\mathbfR^4\) with four critical points are standard, Invent. Math. 79 (1985) 125 · Zbl 0559.57019 |

[18] | K Tanaka, Crossing changes for pseudo-ribbon surface-knots, Osaka J. Math. 41 (2004) 877 · Zbl 1073.57015 |

[19] | K Tanaka, The braid index of surface-knots and quandle colorings, Illinois J. Math. 49 (2005) 517 · Zbl 1077.57022 |

[20] | R Thom, Les singularités des applications différentiables, Ann. Inst. Fourier, Grenoble 6 (1955-1956) 43 · Zbl 0075.32104 |

[21] | O J Viro, Local knotting of sub-manifolds, Mat. Sb. (N.S.) 90(132) (1973) 173, 325 |

[22] | M Yamamoto, Lifting a generic map from a surface into the plane to an embedding into \(4\)-space · Zbl 1154.57027 |

[23] | T Yasuda, Crossing and base numbers of ribbon 2-knots, J. Knot Theory Ramifications 10 (2001) 999 · Zbl 1002.57054 |

[24] | E C Zeeman, Twisting spun knots, Trans. Amer. Math. Soc. 115 (1965) 471 · Zbl 0134.42902 |

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