This represents a surface embedded in four-dimensional space. The sequence of link diagrams is formed by a finite sequence of Reidemeister moves. We.
Marked diagrams and triple points Tsukasa Yashiro Abstract
1
Introduction
A normal form [5] is a sequence of links in 3-sapce with critical bands/discs along with the time axis. A marked diagram is a presentation of a surface in 4-space with a sequence of link diagrams and a link diagram with some crossings with bars (marks). This diagram is obtained from the normal form. This represents a surface embedded in four-dimensional space. The sequence of link diagrams is formed by a finite sequence of Reidemeister moves. We consider a special projection which gives a surface-knot diagram from the normal form. A triple point is generated by the Reidemeister move R-3. In this paper we will estimate the number of the third Reidemeister moves.
2
Reidmeister Moves
Every pair of equivalent knots are deformed into each other by a finite sequence of Reidemeister moves:
R-1
R-2
R-3
Figure 1: Reidemeister moves 1
The Euclidean 4-space R4 is described by R3 × R denoted by R(−∞, ∞) which is the union of the sets R3 × {t} t ∈ (−∞, ∞). For each t ∈ R, R3 [t] is the 3-space. We define the projection pt : R3 (−∞, ∞) → R2 (−∞, ∞) by pt (x, y, z, t) = (x, y, t). We also define the projection proj : R3 (−∞, ∞) → R2 (−∞, ∞) by proj(x, y, z, t) = (x, y, t). Let F be a surface-knot in R4 in a normal form in R3 [a, b]. Let ∆ be the projected image of F under proj. Then [ ∆= pt (F ∩ R3 [t]) t∈[a,b]
Example 2.1. The diagram in the Figure 2 gives a surface-knot diagram without any triple points as the diagram need only Reidemeister move R-2.
D
L− (D)
L+ (D)
Figure 2: A marked graph diagram with three marks from [6].
References [1] J. S. Carter and M. Saito, Knotted surfaces and their diagrams. Mathematical Surveys and Monographs, 55. American Mathematical Society, Providence, RI, 1998. [2] F. Hosokawa, A. Kawauchi, “Proposals for unknotted surfaces in fourspaces”, Osaka J. Math. 16 (1979), 233-248. [3] T. Homma, “Elementary deformations on Homotopy 3Spheres”,Topology and Computer Science, 21-27, Kinokuniya, Tokyo, 1987. 2
[4] A. Kawauchi, “On pseudo-ribbon surface-links”, J. Knot Theory and its Ram. 11, No. 7 (2002) 1043-1062. [5] A. Kawauchi, T. Shibuya and S. Suzuki, Descriptions on surfaces in four-space, I Normal Forms, Math. Sem. Notes Kobe Univ. 10 (1982) 75-125. [6] Jieon Kim, Yewon Joung, and Sang Youl Lee, On generating sets of Yoshikawa moves for marked graph diagrams of suface-links, J. Knot Theory Ramifications 24, 1550018 (2015) (21 pages). [7] D. Roseman, “Reidemeister-type moves for surfaces in four dimensional space”, Banach Center Publications 42 (1998), Knot Theory, 347-380.
3
