Ribbon-clasp surface-links and normal forms of immersed surface-links

Abstract

We introduce the notion of a ribbon-clasp surfacelink, which is a generalization of a ribbon surface-link. We generalize the notion of a normal form on embedded surface-links to the case of immersed surface-links and prove that any (immersed) surface-link can be described in a normal form. It is known that an embedded surface-link is a ribbon surface-link if and only if it can be described in a symmetric normal form. We prove that an (immersed) surface-link is a ribbon-clasp surface-link if and only if it can be described in a symmetric normal form. We also introduce the notion of a ribbon-clasp normal form, which is a simpler version of a symmetric normal form.2000 Mathematics Subject Classification. 57M25, 57M27, 57Q45.

Presentation of immersed surface-links by marked graph diagrams

“…By an argument in [4,5] it is seen that the ambient isotopy class of the immersed surface-link S(D) is uniquely determined from D. We call the immersed surface-link S(D) the immersed surface-link constructed from D. Theorem 2.2. Let L be an immersed surface-link.…”

“…(1) All critical points of L, except the double points, with respect to the projection R 4 = R 3 × R → R are elementary critical points, that are maximal points, saddle points and minimal points. [4] states that any immersed and oriented surfacelink is ambient isotopic to an immersed surface-link satisfying a certain condition. Applying the argument in [5], we can obtain an immersed surface-link required in Theorem 2.2.…”

“…A link is called H-trivial if it is a split union of trivial knots and Hopf links [4]. A trivial link is regarded as an H-trivial link without Hopf links.…”

“…, which is an immersed surface-link in R 4 , where L and L ′ be upper and lower resolutions of D, respectively.…”

On the group of ring motions of an H-trivial link

“…When m = 0, it is a trivial link with n components. H-trivial links are a generalization of trivial links, and play an important role in normal forms of immersed surface-links in R 4 [KK17,KKKL17].…”