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![Labeled crossing diagram and its reflection Example 3.6. Consider the open string α described by the Gauss diagram in Figure 2. Let β be the product α·α * . As in Example 3.2, the slice obstruction in [T04] vanishes. We apply Proposition 3.3 to show that β is not ribbon. We have φ(β)(u, v) = φ(α)(u, v)·φ(α * )(u, v) = φ(α)(u, v)·φ(α)(u −1 , v −1 ). One easily computes that φ(α)(u, v) = −uv 3 −u 3 v 2 +u 3 v 3 +u+v 2 , and from this that φ(β)(u, v) is not equal to φ(β)(v, u). Hence γ is not ribbon. Remark 3.7. The abelian invariant φ cannot detect non-commutativity of open strings, and so it is not effective for the construction in Example 3.2.](profile/Daniel-Silver-2/publication/2114746/figure/fig4/AS:341476639821846@1458425783985/Labeled-crossing-diagram-and-its-reflection-Example-36-Consider-the-open-string-a.png)
Labeled crossing diagram and its reflection Example 3.6. Consider the open string α described by the Gauss diagram in Figure 2. Let β be the product α·α * . As in Example 3.2, the slice obstruction in [T04] vanishes. We apply Proposition 3.3 to show that β is not ribbon. We have φ(β)(u, v) = φ(α)(u, v)·φ(α * )(u, v) = φ(α)(u, v)·φ(α)(u −1 , v −1 ). One easily computes that φ(α)(u, v) = −uv 3 −u 3 v 2 +u 3 v 3 +u+v 2 , and from this that φ(β)(u, v) is not equal to φ(β)(v, u). Hence γ is not ribbon. Remark 3.7. The abelian invariant φ cannot detect non-commutativity of open strings, and so it is not effective for the construction in Example 3.2.
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![Figure 2: A open virtual string α and its Gauss diagram In [SW04] we...](profile/Daniel-Silver-2/publication/2114746/figure/fig7/AS:667078264627208@1536055260812/A-open-virtual-string-a-and-its-Gauss-diagram-In-SW04-we-adapted-Alexander-group_Q320.jpg)
Extended Alexander groups are used to define an invariant for open virtual strings. Examples of non-commuting open strings and a ribbon-concordance obstruction are given. An example is given of a slice virtual open string that is not ribbon. Definitions are extended to open n-strings.
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... arc labeling of D * is obtained from that of D by the involution x → x * . To see this, it suffices to check that if the arc labelings in Figure 9a An open virtual n-string (or simply open n-string) is the image α of a generic immersion R × {1, . . . , n} → R 2 such that (x, j) → (x, j) for sufficiently large negative x, and (x, j) → (x, π j ) for sufficiently large positive x. ...
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Citations
... For further work, we intend to examine the results of Petit [30], who generalized the finite type invariants of Henrich [19], including her universal invariant, and based his generalization on the concept of the matrix for a virtual string, first introduced by Turaev [36,37] and Silver & Williams [33]. ...
We extend the theory of Vassiliev (or finite type) invariants for knots to knotoids using two different approaches. Firstly, we take closures on knotoids to obtain knots and we use the Vassiliev invariants for knots, proving that these are knotoid isotopy invariant. Secondly, we define finite type invariants directly on knotoids, by extending knotoid invariants to singular knotoid invariants via the Vassiliev skein relation. Then, for spherical knotoids we show that there are non-trivial type-1 invariants, in contrast with classical knot theory where type-1 invariants vanish. We give a complete theory of type-1 invariants for spherical knotoids, by classifying linear chord diagrams of order one, and we present examples arising from the affine index polynomial and the extended bracket polynomial.
... Note that the homotopy theory of nanowords over AO can be identified to the theory of open virtual strings which was defined by Turaev in [18]. See also [7] and [14]. ...
... Virtual crossings are not changed by this construction. The idea of using the ascending diagram to define invariants of long flat virtuals is exploited in [47]. The following result is new. ...
This paper is an introduction to virtual knot theory and an exposition of new
ideas and constructions, including the parity bracket polynomial, the arrow
polynomial, the parity arrow polynomial and categorifications of the arrow
polynomial. The paper is relatively self-contained and it describes virtual
knot theory both combinatorially and in terms of the knot theory in thickened
surfaces. The arrow polynomial (of Dye and Kauffman) is a natural
generalization of the Jones polynomial, obtained by using the oriented
structure of diagrams in the state sum. The paper discusses uses of parity
pioneered by Vassily Manturov and uses his parity bracket polynomial to give a
counterexample to a conjecture of Fenn, Kauffman and Manturov. The paper gives
an exposition of the categorification of the arrow polynomial due to Dye,
Kauffman and Manturov and it gives one example (from many found by Aaron
Kaestner) of a pair of virtual knots that are not distinguished by Khovanov
homology (mod 2), or by the arrow polynomial, but are distinguished by a
categorification of the arrow polynomial. Other examples of parity calculations
are indicated. For example, this same pair is distinguished by the parity
bracket polynomial.
... Note that the homotopy theory of nanowords over α 0 can be identified to the theory of open virtual strings which was defined by Turaev in [18]. See also [7] and [14]. ...
This paper is a survey on the recent study of homotopy theory of generalized phrases in Turaev’s theory of words and phrases. In this paper, we introduce the homotopy classification of generalized phrases with some conditions on the numbers of letters. The theory of topology of words and phrases is closely related to the theory of surface-curves. We also introduce applications of the topology of words to the topology of surface-curves.
... The theory of stable equivalence of curves is closely related to the theory of virtual strings. See [8] and [12] for more details.Figure 1. The flattened Reidemeister moves. ...
V. Turaev introduced the theory of topology of words and phrases in 2005. This is a combinatorialy extension of the theory of virtual knots and links. In this paper we generalize the notion of homotopy of words and phrases and we give geometric meanings of the generalized homotopy of words. Moreover using the generalized homotopy theory of words and phrases, we extend some homotopy invariants of nanophrases to $S$-homotopy invariant of nanowords with some homotopy data $S$. Comment: 21 pages, 1 figure; We modified the definition of some invariants. See Remark 5.3
... Remark 4.1. In [5] , D. S. Silver and S. G. Williams studied open virtual multistrings . The theory of open virtual multi-strings is equivalent to the theory of pointed multi-component surface-curves. ...
In this paper we give the stable classification of ordered, pointed, oriented multi-component curves on surfaces with minimal crossing number less than or equal to 2 such that any equivalent curve has no simply closed curves in its components. To do this, we use the theory of words and phrases which was introduced by V. Turaev. Indeed we give the homotopy classification of nanophrases with less than or equal to 4 letters. It is an extension of the classification of nanophrases of length 2 with less than or equal to 4 letters which was given by the author in a previous paper. This is a corrected version of Hokkaido University Preprint Series in Mathematics #921. I corrected the subsection 5.3 and added proofs of propositions.
... Virtual crossings are not changed by this construction. The idea of using the ascending diagram to define invariants of long flat virtuals is exploited in [39]. The following result is new. ...
... We leave the details of this approach to the reader. Figures 36,37,38,39,40 and 41 exhibit the calculation of the extended bracket for the Kishino diagram. Figure 36 is a list of all the states of the Kishino diagram. ...
This paper defines a new invariant of virtual knots and links that we call the extended bracket polynomial, and denote by < > for a virtual knot or link K. This invariant is a state summation over bracket states of the oriented diagram for K. Each state is reduced to a virtual 4-regular graph in the plane and the polynomial takes values in the module generated by these reduced graphs over the ring Q[A,A^{-1}]. The paper is relatively self-contained, with background information about virtual knots and long virtual knots. We give numerous examples applying the extended bracket, including a new proof of the non-triviality of the Kishino diagram and the flat Kishino diagram and non-classicality of single crossing virtualizations. The paper has a section on the estimation of virtual crossing number using the extended bracket state sum. Examples are given of virtual knots with arbitrary minimal embedding genus and arbitrarily high positive difference between the virtual crossing number and the minimal embedding genus. A simplification of < > is introduced and denoted by A[K]. This simplified extended bracket, the arrow polynomial, is a polynomial in an infinite set of variables. It is quite strong (detecting the flat Kishino diagram for example) and easily computable. The paper contains a description of a computer algorithm for A[K] and uses the arrow polynomial, in conjunction with the extended bracket polynomial to determine the minimum genus surfaces on which some virtual knots can be represented.
... This will lead us to a homotopy invariant λ of nanowords taking values in the ring Λ. This invariant is a generalization of an invariant introduced by Silver and Williams [SW2] for curves. Consider the Λ-module K β (w) = K α (w) associated above with a nanoword w : n → A. Let x 0 , x 1 , . . . ...
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... In the setting of curves, this sequence was introduced by Silver and Williams [SW2]. ...
We discuss a topological approach to words introduced by the author. Words on an arbitrary alphabet are approximated by Gauss words and then studied up to natural modifications inspired by the Reidemeister moves on knot diagrams. This leads us to a notion of homotopy for words. We introduce several homotopy invariants of words and give a homotopy classification of words of length five.
We introduce Turaev’s homotopy theory of words and phrases. As new results, we give the classification of oriented ordered pointed irreducible multi-component curves on surfaces which is called monoliteral type with at most six crossings up to stably equivalence using Turaev’s homotopy theory of words and phrases. Moreover we also give the classification of (oriented) ordered pointed irreducible free links of monoliteral type with at most six crossings.
