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Concordance crosscap number of a knot
Author(s) -
Zhang Gengyu
Publication year - 2007
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdm058
Subject(s) - knot (papermaking) , betti number , mathematics , combinatorics , bounding overwatch , artificial intelligence , computer science , chemical engineering , engineering
Abstract We define the concordance crosscap number γ c ( K ) of a knot K as the minimum crosscap number among all the knots concordant to K . The four‐dimensional crosscap number γ *( K ) is the minimum first Betti number of non‐orientable surfaces smoothly embedded in the four‐dimensional ball, bounding the knot K . Clearly, γ *( K ) ⩽ γ c ( K ). We construct two infinite sequences of knots for which γ* ( K ) < γ c ( K ). In particular, the knot 7 4 is one of the examples.