## Algebraic & Geometric Topology

### Widths of surface knots

Yasushi Takeda

#### Abstract

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.

#### Article information

Source
Algebr. Geom. Topol., Volume 6, Number 4 (2006), 1831-1861.

Dates
Revised: 10 August 2006
Accepted: 21 August 2006
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796607

Digital Object Identifier
doi:10.2140/agt.2006.6.1831

Mathematical Reviews number (MathSciNet)
MR2263051

Zentralblatt MATH identifier
1132.57021

#### Citation

Takeda, Yasushi. Widths of surface knots. Algebr. Geom. Topol. 6 (2006), no. 4, 1831--1861. doi:10.2140/agt.2006.6.1831. https://projecteuclid.org/euclid.agt/1513796607

#### References

• P M Akhmet'ev, Smooth immersions of manifolds of small dimension, Mat. Sb. 185 (1994) 3–26
• S Bleiler, M Scharlemann, A projective plane in $\mathbf{R}\sp 4$ with three critical points is standard. Strongly invertible knots have property $P$, Topology 27 (1988) 519–540
• V Carrara, J S Carter, M Saito, Singularities of the projections of surfaces in 4-space, Pacific J. Math. 199 (2001) 21–40
• 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) 265–287
• J S Carter, M Saito, Knotted surfaces and their diagrams, Mathematical Surveys and Monographs 55, American Mathematical Society (1998)
• T Cochran, Ribbon knots in $S\sp{4}$, J. London Math. Soc. $(2)$ 28 (1983) 563–576
• T Fukuda, Topology of folds, cusps and Morin singularities, from: “A fête of topology”, Academic Press, Boston (1988) 331–353
• D Gabai, Foliations and the topology of $3$-manifolds. III, J. Differential Geom. 26 (1987) 479–536
• M Golubitsky, V Guillemin, Stable mappings and their singularities, Graduate Texts in Mathematics 14, Springer, New York (1973)
• F Hosokawa, A Kawauchi, Proposals for unknotted surfaces in four-spaces, Osaka J. Math. 16 (1979) 233–248
• S Kamada, Surfaces in $\mathbf{R}\sp 4$ of braid index three are ribbon, J. Knot Theory Ramifications 1 (1992) 137–160
• S Kamada, Braid and knot theory in dimension four, Mathematical Surveys and Monographs 95, American Mathematical Society (2002)
• A Kawauchi, On pseudo-ribbon surface-links, J. Knot Theory Ramifications 11 (2002) 1043–1062
• H I Levine, 0208609
• J N Mather, 0362393
• O Saeki, Y Takeda, Canceling branch points and cusps on projections of knotted surfaces in 4-space, Proc. Amer. Math. Soc. 132 (2004) 3097–3101
• M Scharlemann, Smooth spheres in $\mathbf{R}\sp 4$ with four critical points are standard, Invent. Math. 79 (1985) 125–141
• K Tanaka, Crossing changes for pseudo-ribbon surface-knots, Osaka J. Math. 41 (2004) 877–890
• K Tanaka, The braid index of surface-knots and quandle colorings, Illinois J. Math. 49 (2005) 517–522
• R Thom, Les singularités des applications différentiables, Ann. Inst. Fourier, Grenoble 6 (1955–1956) 43–87
• O J Viro, Local knotting of sub-manifolds, Mat. Sb. (N.S.) 90(132) (1973) 173–183, 325 (Russian) English translation: Math. USSR-Sb. 19 (1973) 166–176 (1974)
• M Yamamoto, Lifting a generic map from a surface into the plane to an embedding into $4$-space, preprint (2005)
• T Yasuda, Crossing and base numbers of ribbon 2-knots, J. Knot Theory Ramifications 10 (2001) 999–1003
• E C Zeeman, 0195085