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A topological interpretation of Viro's $\mathfrak{gl}(1\vert 1)$-Alexander polynomial of a graph
Geometric Topology · Submitted 19 Jan 2018
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim For an oriented trivalent graph $G$ without source or sink embedded in $S^3$, we prove that the $\mathfrak{gl}(1| 1)$-Alexander polynomial $\underline{\Delta}(G, c)$ defined by Viro satisfies a series of relations, which we call MOY-type relations in [arXiv:1708.09092v2].
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The abstract, sentence by sentence
This is a sequel to [arXiv:1708.09092v2]. For an oriented trivalent graph $G$ without source or sink embedded in $S^3$, we prove that the $\mathfrak{gl}(1| 1)$-Alexander polynomial $\underline{\Delta}(G, c)$ defined by Viro satisfies a series of relations, which we call MOY-type relations in [arXiv:1708.09092v2]. As a corollary we show that the Alexander polynomial $\Delta_{(G, c)}(t)$ studied in [arXiv:1708.09092v2] coincides with $\underline{\Delta}(G, c)$ for a positive coloring $c$ of $G$, where $\Delta_{(G, c)}(t)$ is constructed from certain regular covering space of the complement of $G$ in $S^3$ and it is the Euler characteristic of the Heegaard Floer homology of $G$ that we studied before. When $G$ is a plane graph, we provide a topological interpretation to the vertex state sum of $\underline{\Delta}(G, c)$ by considering a special Heegaard diagram of $G$ and the Fox calculus on the Heegaard surface.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | This is a sequel to [arXiv:1708.09092v2]. | background 1.00 | 0.09 | 0.01 | asserted 0.93 |
| s1 | For an oriented trivalent graph $G$ without source or sink embedded in $S^3$, we prove that the $\mathfrak{gl}(1| 1)$-Alexander polynomial $\underline{\Delta}(G, c)$ defined by Viro satisfies a series of relations, which we call MOY-type relations in [arXiv:1708.09092v2]. | result 0.65 | 0.89 | 0.66 | asserted 1.00 |
| s2 | As a corollary we show that the Alexander polynomial $\Delta_{(G, c)}(t)$ studied in [arXiv:1708.09092v2] coincides with $\underline{\Delta}(G, c)$ for a positive coloring $c$ of $G$, where $\Delta_{(G, c)}(t)$ is constructed from certain regular covering space of the complement of $G$ in $S^3$ and it is the Euler characteristic of the Heegaard Floer homology of $G$ that we studied before. | result 0.99 | 0.85 | 0.38 | asserted 1.00 |
| s3 | When $G$ is a plane graph, we provide a topological interpretation to the vertex state sum of $\underline{\Delta}(G, c)$ by considering a special Heegaard diagram of $G$ and the Fox calculus on the Heegaard surface. | contribution 0.44 | 0.94 | 0.20 | asserted 1.00 |
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Original abstract
This is a sequel to [arXiv:1708.09092v2]. For an oriented trivalent graph $G$ without source or sink embedded in $S^3$, we prove that the $\mathfrak{gl}(1| 1)$-Alexander polynomial $\underline{\Delta}(G, c)$ defined by Viro satisfies a series of relations, which we call MOY-type relations in [arXiv:1708.09092v2]. As a corollary we show that the Alexander polynomial $\Delta_{(G, c)}(t)$ studied in [arXiv:1708.09092v2] coincides with $\underline{\Delta}(G, c)$ for a positive coloring $c$ of $G$, where $\Delta_{(G, c)}(t)$ is constructed from certain regular covering space of the complement of $G$ in $S^3$ and it is the Euler characteristic of the Heegaard Floer homology of $G$ that we studied before. When $G$ is a plane graph, we provide a topological interpretation to the vertex state sum of $\underline{\Delta}(G, c)$ by considering a special Heegaard diagram of $G$ and the Fox calculus on the Heegaard surface.
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1 arXiv paper cite this in 1 sentence
What it cites · 11 sentences without a confident label
Builds on 1 paper
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In this section, we discuss a series of relations that $Δ(G, c)$ satisfies, which are adapted versions of the MOY-type relations in [ref] .
An Alexander polynomial for MOY graphs · MOY-type relations for $Δ(G, c)$. · 0.80
Uses the method or tool of 1 paper
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From the Heegaard diagram $H$ we can calculate $Δ_(G, c)(t)$ by applying Fox calculus as we discussed in [ref] .
Floer homology and embedded bipartite graphs · A vertex state sum formula for $Δ_(G, c)(t)$ · 1.00
Compares with 1 paper
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Comparing the relations in [ref] with those in Theorem [ref], it is easy to see that both sides of the equation above satisfy exactly the same MOY-type relations.
An Alexander polynomial for MOY graphs · Relations between $Δ_(G, c)(t)$ and $Δ(G, c)$ · 0.92
Cites as support 1 paper
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The relation follows from [ref] , the definition of $Δ_(G, c)(t)$.
An Alexander polynomial for MOY graphs · A vertex state sum formula for $Δ_(G, c)(t)$ · 0.96
Cites as background 1 paper
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For more details and properties of it, please refer to [ref] .
An Alexander polynomial for MOY graphs · Relations between $Δ_(G, c)(t)$ and $Δ(G, c)$ · 0.99
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