Minimal generating sets of Reidemeister moves

Polyak, Michael

doi:10.4171/qt/10

Cited by 127 publications

(107 citation statements)

References 12 publications

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“…As the canonical source-sink orientations are dependant on the orientation of the crossings we must show invariance under the oriented Reidemeister moves. In order to minimize the number of changes in cut loci in the diagrams outside of where the Reidemeister move occurs we chose to work with a set of oriented moves shown by Polyak [38], Theorem 1.2, to be minimal.…”

Section: Proof Of Invariancementioning

confidence: 99%

Khovanov homology, Lee homology and a Rasmussen invariant for virtual knots

Dye

Kaestner

Kauffman

2017

J. Knot Theory Ramifications

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The paper contains an essentially self-contained treatment of Khovanov homology, Khovanov-Lee homology as well as the Rasmussen invariant for virtual knots and virtual knot cobordisms which directly applies as well to classical knot and classical knot cobordisms. We give an alternate formulation for the Manturov definition [34] of Khovanov homology [25] [26] for virtual knots and links with arbitrary coefficients. This approach uses cut loci on the knot diagram to induce a conjugation operator in the Frobenius algebra. We use this to show that a large class of virtual knots with unit Jones polynomial are nonclassical, proving a conjecture in [20] and [10]. We then discuss the implications of the maps induced in the aforementioned theory to the universal Frobenius algebra [27] for virtual knots. Next we show how one can apply the Karoubi envelope approach of Bar-Natan and Morrison [3] on abstract link diagrams [17] with cross cuts to construct the canonical generators of the Khovanov-Lee homology [30]. Using these canonical generators we derive a generalization of the Rasmussen invariant [39] for virtual knot cobordisms and generalize Rasmussen's result on the slice genus for positive knots to the case of positive virtual knots. It should also be noted that this generalization of the Rasmussen invariant provides an easy to compute obstruction to knot cobordisms in S g × I × I in the sense of Turaev [42].

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Section: Proof Of Invariancementioning

confidence: 99%

Khovanov homology, Lee homology and a Rasmussen invariant for virtual knots

Dye

Kaestner

Kauffman

2017

J. Knot Theory Ramifications

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“…In the same spirit as small "sets" of Reidemeister moves can generate all Reidemeister moves (as is shown in [18]), one naturally expects some 2-meridians to express as linear combinations of others. Since the equations derived from are by far the most numerous and the most complicated, we try to get rid of them in priority.…”

Section: Keeping Only One Tetrahedron Equationmentioning

confidence: 95%

Finite-type 1-cocycles of knots given by Polyak–Viro Formulas

Mortier

2015

J. Knot Theory Ramifications

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We present a new method to produce simple formulas for 1-cocycles of knots over the integers, inspired by Polyak-Viro's formulas for finite-type knot invariants. We conjecture that these 1-cocycles represent finite-type cohomology classes in the sense of Vassiliev. An example of degree 3 is studied, and shown to coincide over Z 2 with the Teiblum-Turchin cocycle v 1 3 .The study of the topology of the space of knots was initiated in 1990 by Vassiliev [23], who defined finite-type cohomology classes by applying ideas from the finitedimensional affine theory of plane arrangements to the infinite-dimensional theory of long knots in 3-space. Here, of finite type means of finite complexity, in some sense, hence hopefully computable. Independently, outstanding general results on the topology of knot spaces have been obtained by Hatcher [11], leading to the idea that higher dimensional invariants of knots -in particular, 1-cocycles -should capture information about the geometry of a knot (see [7]). The zeroth level of Vassiliev's theory, known as finite-type knot invariants, has been extensively studied in the subsequent years [1,2,9,13]. However at the first level -that of 1-cocycles -only one example, in degree 3, has been proved to exist by Teiblum and Turchin, and then actually described by Vassiliev with a formula over Z 2 [22,25]. Since then, no progress has been made, probably because of the technicity of Vassiliev's construction and the apparent difficulty of turning it into a systematic method: indeed, it involves singularity theory with differential geometric 1540004-1 J. Knot Theory Ramifications 2015.24. Downloaded from www.worldscientific.com by CALIFORNIA INSTITUTE OF TECHNOLOGY on 06/27/16. For personal use only.

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“…The commutativity of the diagrams (10) and (11) means that (ϕ α , (ϕ α ) 2 , (ϕ α ) 0 ) is a monoidal endofunctor of C. The diagrams (12) and (13) indicate that the natural transformation ϕ 2 (α, β) is monoidal. The diagram (14) and the equality (ϕ 0 ) ½ = (ϕ 1 ) 0 indicate that the natural transformation ϕ 0 is monoidal.…”

Section: G-crossed Categoriesmentioning

confidence: 98%

“…Note that the type 3 moves determined by various orientations of the branches may be expanded as compositions of the type 3 move of Fig. 9 and the type 2 moves, see for instance [12]. Therefore, it is enough to consider only the type 3 move shown in Fig.…”

Section: Presentation Of Graphs By Diagramsmentioning

confidence: 98%