TY - JOUR
T1 - Surface links and their generic planar projections
AU - Saeki, Osamu
AU - Takeda, Yasushi
N1 - Funding Information:
The first named author has been supported in part by Grant-in-Aid for Scientific Research (No. 19340018), Japan Society for the Promotion of Science. The second named author has been supported by 21st century COE program “Development of Dynamic Mathematics with High Functionality” and by the GP program at Faculty of Mathematics, Kyushu University, Japan.
Copyright:
Copyright 2009 Elsevier B.V., All rights reserved.
PY - 2009/1
Y1 - 2009/1
N2 - We often study surface links in 4-space by using their projections into 3-space, or their broken surface diagrams. It is well-known that a broken surface diagram recovers the given surface link. In this paper, we study surface links in 4-space by using their generic projections into the plane. These projections have fold points and cusps as their singularities in general. We consider the question whether such a generic planar projection can recover the given surface link. We introduce the notion of banded braids, and show that a generic planar projection together with banded braids associated to the segments of the fold curve image can recover the given surface link. As an application, we give a new proof to the Whitney congruence concerning the normal Euler number of surface links.
AB - We often study surface links in 4-space by using their projections into 3-space, or their broken surface diagrams. It is well-known that a broken surface diagram recovers the given surface link. In this paper, we study surface links in 4-space by using their generic projections into the plane. These projections have fold points and cusps as their singularities in general. We consider the question whether such a generic planar projection can recover the given surface link. We introduce the notion of banded braids, and show that a generic planar projection together with banded braids associated to the segments of the fold curve image can recover the given surface link. As an application, we give a new proof to the Whitney congruence concerning the normal Euler number of surface links.
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U2 - 10.1142/S0218216509006847
DO - 10.1142/S0218216509006847
M3 - Article
AN - SCOPUS:65249133003
SN - 0218-2165
VL - 18
SP - 41
EP - 66
JO - Journal of Knot Theory and its Ramifications
JF - Journal of Knot Theory and its Ramifications
IS - 1
ER -