A self-linking invariant of virtual knots

By Louis H. Kauffman. 2004

91 citations recorded by the source corpus.

In this paper we introduce a new invariant of virtual knots and links that is non-trivial for many virtuals, but is trivial on classical knots and links. The invariant will initially be expressed in terms of a relative of the bracket polynomial [4], and then extracted from this polynomial in terms of its exponents, particularly for the case of knots. The analog of the bracket polynomial will be denoted {K} (with curly brackets) and called the binary bracket polynomial. See Section 3 for the definition and properties of the binary bracket. The key to the combinatorics of this invariant is an interpretation of the state sum in terms of 2-colorings of the associated diagrams.

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

View Paper: https://www.semanticscholar.org/paper/ed7a1ca1c33c888cff9c87f1e1226f2cb1d29942

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

A polynomial invariant of virtual links (2013, methodology): “Following [11] we use Odd ( K ) to denote all the odd crossings of K .”
Parity in knot theory (2010, background): “Kauffman was the first to use odd crossings (odd chords), to construct invariants of virtual knots: he defined the odd self-linking index for virtual knots in [2]; this index vanishes on classical knots.”
Virtual knot theory-unsolved problems (2005, unknown): “…[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] [94] [95] [96] [97] [98] [99]…”
A polynomial invariant of virtual knots (2012, methodology): “Given a virtual knot diagram K, similar to [3] we use Odd(K) to denote all the odd crossings of it.”
An Extended Bracket Polynomial for Virtual Knots and Links (2007, background): “The odd writhe [23] is the sum of the signs of the odd crossings.”