The generalized Yamada polynomials of virtual spatial graphs

Deng, Qingying; X, Jin; Kauffman, Louis H.

doi:10.1016/j.topol.2019.01.003

Cited by 6 publications

(9 citation statements)

References 28 publications

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“…Arguably, one of the most utilized polynomials to study classical spatial graphs is the Yamada polynomial. There are three polynomials for virtual spatial graphs related to the Yamada polynomial that one can turn into graphoid invariants [8,10,20]. We will focus mainly on Fleming-Mellor's formulation R(G) and Deng-Jin-Kauffman's formulation R(G; A, x).…”

Section: Polynomial Invariantsmentioning

confidence: 99%

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Graphoids

Gügümcü¹,

Kauffman²,

Pongtanapaisan³

2022

Preprint

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We study invariants of virtual graphoids, which are virtual spatial graph diagrams with two distinguished degree-one vertices modulo graph Reidemeister moves applied away from the distinguished vertices. Generalizing previously known results, we give topological interpretations of graphoids. There are several applications to virtual graphoid theory. First, virtual graphoids are suitable objects for studying knotted graphs with open ends arising in proteins. Second, a virtual graphoid can be thought of as a way to represent a virtual spatial graph without using as many crossings, which can be advantageous for computing invariants.

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Section: Polynomial Invariantsmentioning

confidence: 99%

“…On the other hand, there is an upper estimate for spanR(G) from Lemma 5. Deng, Kauffman, and Jin developed a Yamada-type polynomial denoted by R(G; A, 1) for virtual spatial graphs [8]. In this section, we will use R(G; A, 1) to show that certain virtual graphoids cannot be presented as a diagram without virtual crossings.…”

Section: 1mentioning

confidence: 99%

Graphoids

Gügümcü¹,

Kauffman²,

Pongtanapaisan³

2022

Preprint

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“…Corresponding to marked Reidemeister moves in marked Gauss diagrams we have defined moves for marked virtual link diagrams and there is one-to-one correspondence between marked Gauss diagrams and marked virtual link diagrams under the equivalence relation generated by moves in the corresponding sets. In [17], Q. Deng, X. Jin, and L. Kauffman gave a topological interpretation of virtual spatial graph diagrams. It would be interesting to know a topological interpretation of marked virtual link diagrams.…”

Section: Introductionmentioning

confidence: 99%

“…For various aspects in spatial graph theory and virtual spatial graph theory we refer the reader to papers [8,26,31,33,36,41,45,52,53] and [17,19,20,35,44], respectively.…”

Section: Introductionmentioning

confidence: 99%

Virtually Symmetric representations and marked Gauss diagrams

Bardakov,

Neshchadim,

Singh

2020

Preprint

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In this paper we define virtually symmetric representations of virtual braid group V Bn and prove that many previously known representations are equivalent to virtually symmetric representations. Using a virtually symmetric representation we define virtual link group. We define and study group system of virtual knots by defining marked Gauss diagrams as an extension of Gauss diagrams. A marked Gauss diagram is a Gauss diagram with marked nodes (signed vertices) on circles, and are not attached to arrows. In particular, we extend the definition of virtual link group to marked Gauss diagrams and define peripheral structure for 1-circle marked Gauss diagrams. We define Cm-groups and prove that every irreducible C1-group can be realized as the group of a marked Gauss diagram. We give an interpretation of marked Gauss diagrams in terms of virtual spatial graph diagrams with marked nodes. We also study peripherally specified homomorphic image of groups associated to marked Gauss diagrams.

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“…The generalized Yamada polynomials of virtual spatial graphs were recently introduced by Q. Deng, X. Jin and L.H. Kauffnan in [6].…”

Section: Introductionmentioning

confidence: 99%

Density of Roots of the Yamada Polynomial of Spatial Graphs

Lei

et al. 2019

Proc. Steklov Inst. Math.

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We survey the construction and properties of the Yamada polynomial of spatial graphs and present the Yamada polynomial formulae for some classes of graphs. Then we construct an infinite family of spatial graphs for which roots of Yamada polynomials are dense in the complex plane. It is well known due S. Kinoshita [14,15], that the Alexander ideal and Alexander polynomial are invariants of spatial graphs which are determined by the fundamental groups of the complements of spatial graphs.In 1989, S. Yamada [24] introduced Yamada polynomial of spatial graphs in R 3 . It is an concise and useful ambient isotopy invariant for graphs with maximal degree less than four. There are many interesting results on Yamada polynomial and its generalizations. J. Murakami [17] investigated the two-variable extension Z S of the Yamada polynomial and gave an invariant related to the HOMFLY polynomial. In 1994, the crossing number of spatial graphs in terms of the reduced degree of Yamada polynomial has been studied by T. Motohashi, Y. Ohyama and K. Taniyama [19]. In 1996, A. Dobrynin and A. Vesnin [7] studied properties of the Yamada polynomial of spatial graphs. For any graph G, V. Vershinin and A. Vesnin [23] defined bigraded cohomology groups whose graded Euler characteristic is a multiple of the Yamada polynomial of G. Another invariant of spatial graphs associated with U q (sl(2, C)) was introduced 2010 Mathematics Subject Classification. Primary 57M15; Secondary 05C31.

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The generalized Yamada polynomials of virtual spatial graphs

Cited by 6 publications

References 28 publications

Graphoids

Graphoids

Virtually Symmetric representations and marked Gauss diagrams

Density of Roots of the Yamada Polynomial of Spatial Graphs

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