Minimal surface representations of virtual knots and links

By H. Dye, Louis H. Kauffman. 2004

99 citations recorded by the source corpus.

Kuperberg (15) has shown that a virtual knot diagram corre- sponds (up to generalized Reidemeister moves) to a unique embedding in a thickened surface of minimal genus. If a virtual knot diagram is equivalent to a classical knot diagram then this minimal surface is a sphere. Using this result and a generalised bracket polynomial, we develop methods that may determine whether a virtual knot diagram is non-classical (and hence non-trivial). As examples we show that, except for special cases, link dia- grams with a single virtualization and link diagrams with a single virtual crossing are non-classical. AMS Classification 57M25, 57M27; 57N05

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

View Paper: https://www.semanticscholar.org/paper/d8f29378eadf73a9478fa219bf8477adce2356ba

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Virtual Crossing Number and the Arrow Polynomial (2008, background): “Previous results that determine whether a link diagram is virtual or classical include [3], [14], and [15].”
Virtual knot theory-unsolved problems (2005, unknown): “The Kishino diagram has been shown non-trivial by a calculation of the three-strand Jones polynomial [68], by the surface bracket polynomial of Dye and Kauffman [16, 17], by the Ξ-polynomial (the surface generalization of the Jones polynomial of Manturov [82], and its biquandle has been shown to be ”
Knot Diagrammatics (2004, methodology): “The Kishino diagram has been shown non-trivial by a calculation of the three-strand Jones polynomial [85], by the surface bracket polynomial of Dye and Kauffman [25, 26], by the Ξ-polynomial (the surface generalization of the Jones polynomial of Manturov [105], and its biquandle has been shown to be”
VIRTUAL BRAIDS AND THE L-MOVE (2005, background): “For reference to previous work on virtual links and braids the reader should consult [2, 6, 7, 9, 11, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 27, 28, 32, 36, 37].”
An Affine Index Polynomial Invariant of Virtual Knots (2012, background): “See [20, 1, 4, 24] for more information about this point of view.”
Virtual Braids (2004, background): “For reference to previous work on virtual braids the reader should consult [5, 6, 9, 10, 11, 17, 18, 19, 12, 13, 14, 20, 21, 23, 25, 26, 27, 28].”