The Yamada polynomial of spatial graphs obtained by edge replacements

Li, Miaowang; Lei, Fengchun; Li, Fengling; Vesnin, Andrei

doi:10.1142/s021821651842004x

Cited by 8 publications

(6 citation statements)

References 24 publications

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“…(iv) Let Θ s be the graph consisting of two vertices and s edges between them, also known as "s-theta-graph" (see Figure 1). Then The following property of H(G) was obtained in [16].…”

Section: Yamada Polynomial Of a Graphmentioning

confidence: 99%

“…Theorem 2.8. [16] Let G be a connected labelled graph, and G(K E ) be the graph obtained from G by replacing the edge a by a connected graph K a ∈ K E for every edge a of G. If we replace w by −σ, and replace a by γ a for every label a in Ch(G), then we get…”

Section: Definition 25 the Chain Polynomial Ch(g) Of A Labelled Graph...mentioning

confidence: 99%

“…This paper consists of the introduction and three parts. In the first part (Section 2) we recall properties of the Yamada polynomial of graphs and recall some formulae from [16] for computing the Yamada polynomial of graphs by edge replacements via the chain polynomial (see Theorem 2.8). In the second part (Section 3) we discuss formulae for computing the Yamada polynomial of spatial graphs obtained by replacing edges of cycle graphs, theta-graphs, or bouquet graphs by spatial parts (see Theorem 3.7 and [16]).…”

Section: Introductionmentioning

confidence: 99%

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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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“…(iv) Let Θ s be the graph consisting of two vertices and s edges between them, also known as "s-theta-graph" (see Figure 1). Then The following property of H(G) was obtained in [16].…”

Section: Yamada Polynomial Of a Graphmentioning

confidence: 99%

Section: Definition 25 the Chain Polynomial Ch(g) Of A Labelled Graph...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Density of Roots of the Yamada Polynomial of Spatial Graphs

Lei

et al. 2019

Proc. Steklov Inst. Math.

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“…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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“…For related studies of spatial graphs, please refer to [9,10,20,21,23,25]. For related studies of virtual spatial graphs, please refer to [4,5,19,22,24].…”

Section: Introductionmentioning

confidence: 99%

The generalized Yamada polynomials of virtual spatial graphs

Deng

Kauffman

2019

Topology and its Applications

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Classical knot theory can be generalized to virtual knot theory and spatial graph theory. In 2007, Fleming and Mellor combined virtual knot theory and spatial graph theory to form, combinatorially, virtual spatial graph theory. In this paper, we introduce a topological definition of virtual spatial graphs that is similar to the topological definition of a virtual link. Our main goal is to generalize the classical Yamada polynomial that is defined for a spatial graph. We define a generalized Yamada polynomial for a virtual spatial graph and prove that it can be normalized to a rigid vertex isotopic invariant and to a pliable vertex isotopic invariant for graphs with maximum degree at most 3. We consider the connection and difference between the generalized Yamada polynomial and the Dubrovnik polynomial of a classical link. The generalized Yamada polynomial specializes to a version of the Dubrovnik polynomial for virtual links such that it can be used to detect the nonclassicality of some virtual links. We obtain a specialization for the generalized Yamada polynomial (via the Jones-Wenzl projector P 2 acting on a virtual spatial graph diagram), that can be used to write a program for calculating it based on Mathematica Code.2000 Mathematics Subject Classification. 57M27 .

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The Yamada polynomial of spatial graphs obtained by edge replacements

Cited by 8 publications

References 24 publications

Density of Roots of the Yamada Polynomial of Spatial Graphs

Density of Roots of the Yamada Polynomial of Spatial Graphs

Virtually Symmetric representations and marked Gauss diagrams

The generalized Yamada polynomials of virtual spatial graphs

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