Unknotting Virtual Knots With Gauss Diagram Forbidden Moves

Nelson, Sam

doi:10.1142/s0218216501001244

Cited by 81 publications

(60 citation statements)

References 4 publications

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“…We call this the unrestricted virtual braid group, denoted UB n . It is known that any classical knot can be unknotted in the virtual category if we allow both forbidden moves [17,31]. Nevertheless, linking phenomena still remain.…”

Section: Welded Links and Unrestricted Virtualsmentioning

confidence: 99%

Virtual Braids and the L-Move

Kauffman

Lambropoulou

2006

J. Knot Theory Ramifications

103

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In this paper we prove a Markov Theorem for the virtual braid group and for some analogs of this structure. The virtual braid group is the natural companion to the category of virtual knots, just as the Artin braid group is to classical knots and links. In classical knot theory the braid group gives a fundamental algebraic structure associated with knots. The Alexander Theorem tells us that every knot or link can be isotoped to braid form. The capstone of this relationship is the Markov Theorem, giving necessary and sufficient conditions for two braids to close to the same link (where sameness of two links means that they are ambient isotopic).The Markov Theorem in classical knot theory is not easy to prove. The Theorem was originally stated by A.A. Markov with three moves and then N. Weinberg reduced them to the known two moves [29,38]. The first complete proof is due to J. Birman [4]. Other published proofs are due to D. Bennequin [3], H. Morton [30], P. Traczyk [35] and S. Lambropoulou [24,25]. In this paper we shall follow the "L-Move" methods of Lambropoulou. In the L-move approach to the Markov theorem, one gives a very simple uniform move that can be applied anywhere in a braid to produce a braid with the same closure. This move, the L-move, consists in cutting a strand of the braid and taking the top of the cut to the bottom of the braid (entirely above or entirely below the braid) and taking the bottom of the cut to the top of the braid (uniformly above or below in correspondence with the choice for the other end of the cut). See Figure 15 for an illustration of a classical L-move. One then proves that two braids have the same closure if and only if they are related by a sequence of L-moves. Once this L-Move Markov Theorem is established, one can reformulate the result in various ways, including the more algebraic classical Markov Theorem that uses conjugation and stabilization moves to relate braids with equivalent closures.Up to now [25,10,26] the L-moves were only used for proving analogues of the Markov theorem for classical knots and links in 3-manifolds (with or without boundary). Our approach to a Markov Theorem for virtual knots and links follows a similar strategy to the classical case, but necessarily must take into account properties of virtual knots and links that diverge from the classical case. In particlular, we use L-moves that are purely virtual, as well as considering the effect of allowed and forbidden moves of the virtual

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Section: Welded Links and Unrestricted Virtualsmentioning

confidence: 99%

Virtual Braids and the L-Move

Kauffman

Lambropoulou

2006

J. Knot Theory Ramifications

103

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“…Three of the moves involve only classical crossings, three of the moves involve only virtual crossings, and one of the moves involves two virtual crossings and one real crossing. Note that the two obvious 3-crossing moves that involve two real crossings and one virtual crossing are forbidden because including them makes all links are equivalent to unlinks by Nelson [17].…”

Section: Virtual Linksmentioning

confidence: 99%

Twisted link theory

Bourgoin

2008

Algebr. Geom. Topol.

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We introduce stable equivalence classes of oriented links in orientable three-manifolds that are orientation I -bundles over closed but not necessarily orientable surfaces. We call these twisted virtual links and show that they subsume the virtual knots introduced by L Kauffman and the projective links introduced by Yu V Drobotukhina. We show that these links have unique minimal genus three-manifolds. We use link diagrams to define an extension of the Jones polynomial for these links and show that this polynomial fails to distinguish two-colorable links over nonorientable surfaces from non-two-colorable virtual links. 57M25, 57M27, 57M15, 57M05

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“…Virtualizing x but not i(x) = x (or r(x) = x) results in an invalid pseudo-II move with one classical and one virtual crossing. If any crossing in the set of ir classes appears in a type III move with two classical crossings not in the set, virtualizing that crossing will change the III move into an invalid move, either one of the two forbidden moves F t or F h of figure 4 or an invalid move equivalent to one of the two forbidden move sequences F o or F s in [7].…”

Section: Ir Classes and Realization Setsmentioning

confidence: 99%

Virtual Crossing Realization

Nelson

2005

J. Knot Theory Ramifications

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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 virtual isotopy sequences is proposed, and a sufficient condition for realizability is found. The conjecture is reformulated in terms of 2-knots and knots in thickened surfaces.

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Unknotting Virtual Knots With Gauss Diagram Forbidden Moves

Cited by 81 publications

References 4 publications

Virtual Braids and the L-Move

Virtual Braids and the L-Move

Twisted link theory

Virtual Crossing Realization

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