ABSTRACT LINK DIAGRAMS AND VIRTUAL KNOTS

By Naoko Kamada, S. Kamada. 2000

245 citations recorded by the source corpus.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/158563b2177a5e5f88a5545c167d3fa381260898

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

GEOMETRIC INTERPRETATIONS OF QUANDLE HOMOLOGY (2000, methodology): “Abstract knot diagrams are defined in [15] and the relation to virtual knots [16] is established in [15].”
Virtual knot theory-unsolved problems (2005, unknown): “See [2] [3] [4] [5] [9] [10] [11] [12] [8] [13] [16] [17] [18] [19] [20] [22] [23] [24] [25] [26] [31] [33] [34] [36] [37] [38] [39] [40] [41] [43] [50] [51] [55] [58] [59] [60] [61] [62] [64] [65] [67] [68] [70] [72] [73] [74] [75] [76] [78] [79] [80] [81] [82] [84] [86] [87] [88] [83] [90] [93]…”
Homology theory for the set-theoretic Yang–Baxter equation and knot invariants from generalizations of quandles (2002, background): “The fundamental (Wirtinger) groups of virtual knots were studied [22, 28] and their geometric interpretations were given [17].”
An Extended Bracket Polynomial for Virtual Knots and Links (2007, background): “The abstract knot diagram is a re lization of the knot on the left in a thickened surface with boundary and it is obtained by making a neighborhood of the virtual diagram that resolves the virtual crossing into arcs that travel on separate bands.”
Twisted link theory (2006, background): “In [10], Naoko Kamada and Seiichi Kamada showed that the group of a virtual knot is the group of a three-complex obtained by collapsing one of the boundary components of the complement of the link in the thickened surface.”
The rack space (2003, methodology): “However, the proof of this last result spreads over three papers [3, 18, 19], so it seems worthwhile to include here a short direct proof.”
Virtual Braids (2004, background): “Another way to make this representation is to form a ribbon-neighborhood surface (sometimes called an abstract link diagram [14]) for a given virtual knot or link, as illustrated in Figure 6.”
Biquandles and virtual links (2004, background): “Two virtual link diagrams are equivalent if and only if their correspond surface embeddings are stably equivalent, [13,15,17,4] .”
QUATERNIONIC INVARIANTS OF VIRTUAL KNOTS AND LINKS (2006, unknown): “a virtual crossing. The virtual crossing is indicated algebraically by τ i. The group of virtual braids on nstrings is denoted by VB n. Kamada’s presentation of the virtual braid group is as follows, [KK] . Let generators of VB n be σ1, σ2,...,σ n−1 where σ i corresponds to the real positive crossin”
ALEXANDER GROUPS AND VIRTUAL LINKS (2001, background): “(However, see [5] for an interpretation of G as the fundamental group of a singular 3-manifold.”
STABLE EQUIVALENCE OF KNOTS ON SURFACES AND VIRTUAL KNOT COBORDISMS (2000, background): “A bijective relation between virtual knots and certain knots on surfaces, called abstract knots was given [7].”