Up–down colorings of virtual-link diagrams and the necessity of Reidemeister moves of type II
2017 ◽
Vol 26
(12)
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pp. 1750073
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Keyword(s):
Type Ii
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We introduce an up–down coloring of a virtual-link (or classical-link) diagram. The colorabilities give a lower bound of the minimum number of Reidemeister moves of type II which are needed between two [Formula: see text]-component virtual-link (or classical-link) diagrams. By using the notion of a quandle cocycle invariant, we give a method to detect the necessity of Reidemeister moves of type II between two given virtual-knot (or classical-knot) diagrams. As an application, we show that for any virtual-knot diagram [Formula: see text], there exists a diagram [Formula: see text] representing the same virtual-knot such that any sequence of generalized Reidemeister moves between them includes at least one Reidemeister move of type II.
2005 ◽
Vol 14
(08)
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pp. 1045-1075
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Keyword(s):
2017 ◽
Vol 26
(12)
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pp. 1750072
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Keyword(s):
2009 ◽
Vol 18
(11)
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pp. 1577-1596
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Keyword(s):
2013 ◽
Vol 22
(13)
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pp. 1350073
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Keyword(s):
2000 ◽
Vol 09
(01)
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pp. 93-106
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Keyword(s):
2019 ◽
Vol 30
(14)
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pp. 1950072
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Keyword(s):
2003 ◽
Vol 12
(06)
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pp. 781-803
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Keyword(s):
1993 ◽
Vol 02
(03)
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pp. 251-284
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Keyword(s):
2017 ◽
Vol 26
(10)
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pp. 1750051
Keyword(s):
2015 ◽
Vol 24
(02)
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pp. 1550010
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