Figure 5 - uploaded by Sam Nelson
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We study virtual isotopy sequences with classical initial and final diagrams, asking when such a sequence can be changed into a classical isotopy sequence by replacing virtual crossings with classical crossings. An example of a sequence for which no such virtual crossing realization exists is given. A conjecture on conditions for realizability of v...
Context in source publication
Context 1
... knots and links may then be regarded as equivalence classes of Gauss diagrams under the Gauss diagram versions of the Reidemeister moves; a Gauss diagram determines a virtual knot diagram up to virtual moves (vI, vII, vIII and v), while a virtual knot diagram determines a unique Gauss diagram. There are several instances of type III moves depending on the orientation and cyclic order of the three strands involved; only two of these are listed in figure 5. ...
Citations
... Classical knot theory forms a subset of virtual knot theory, since every classical knot may be considered a virtual knot without virtual crossings, and, crucially, virtually isotopic classical knots are classically isotopic [8], [14], [16]. Many invariants of classical knots extend to virtual knots by simply ignoring virtual crossings; these include the knot group, the knot quandle and biquandle, the various skein polynomials, finite type invariants, etc. ...
... Given a Gauss code, one asks whether the Gauss code corresponds to a classical knot. If reconstruction yields a planar knot diagram, then the answer is clearly yes; however, every classical knot has representative Gauss codes which determine non-planar, i.e., virtual, diagrams, [16]. Thus, if a Gauss code is not obviously classical, the code may well represent either a classical or non-classical virtual knot. ...
Non-classical virtual knots may have non-isomorphic upper and lower quandles. We exploit this property to define the quandle difference invariant, which can detect non-classicality by comparing the numbers of homomorphisms into a finite quandle from a virtual knot's upper and lower quandles. The invariants for small-order finite quandles detect non-classicality in several interesting virtual knots. We compute the difference invariant with the six smallest connected quandles for all non-evenly intersticed Gauss codes with 3 and 4 crossings. For non-evenly intersticed Gauss codes with 4 crossings, the difference invariant detects non-classicality in 86% of codes which have non-trivial upper or lower counting invariant values.
... Realizations were originally investigated by Sam Nelson [13]. In this paper, Nelson showed that a sequence of virtual Reidemeister and Reidemeister moves performed on a virtual link diagram could not always be replicated by a sequence of classical Reidemeister moves performed on a realization. ...
A realization of a virtual link diagram is obtained by choosing over/under markings for each virtual crossing. Any realization can also be obtained from some representation of the virtual link. (A representation of a virtual link is a link diagram on an oriented 2-dimensional surface.) We prove that if a minimal genus representation meets certain criteria then there is a minimal genus representation resulting in a knotted realization.
... We would like to take this opportunity to thank the many people who have, at the time of this writing, worked on the theory of virtual knots and links. The ones explicitly mentioned or referenced in this paper are: R. S. Avdeev, V. G. Bardakov [2,3,4,5,9,10,11,12,8,13,16,17,18,19,20,22,23,24,25,26,32,34,35,37,38,39,40,41,42,44,51,52,56,59,60,61,62,63,65,66,68,69,71,73,74,75,76,77,79,80,81,82,83,85,87,88,89,84,91,94,95,96,97,98,99,100,101,102,103,104,105,106,107,110]. We apologize to anyone who was left out of this list of participant researchers, and we hope that the problems described herein will stimulate people on and off this list to enjoy the beauty of virtual knot theory! ...
This paper is an introduction to the theory of virtual knots and links and it gives a list of unsolved problems in this subject.
We classify closed virtual 2-braids completely as virtual links. For the proof, we use surface bracket polynomial due to Dye and Kauffman, Kuperberg's theorem which states existence and uniqueness of a minimal realization for a virtual link, and a subgroup of the 3-braid group.
