Spin networks, Ehrhart quasipolynomials, and combinatorics of dormant indigenous bundles

Wakabayashi, Yasuhiro

doi:10.1215/21562261-2019-0020

Cited by 8 publications

(14 citation statements)

References 16 publications

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“…II], which refers to NNI moves asS-transformations, where the 'S' stands for slide. For the slightly more general case of {1, 3}-graphs, the proof of Theorem 1 provides an alternative proof for a proposition by Wakabayashi [15,Prop. 6.2], which refers to NNI moves as A-moves.…”

Section: Lemma 7 the External Edges Of A Caterpillar Can Be Sorted Amentioning

confidence: 97%

“…To deal with weights on the edges of a {1, 3}-graph, we now enhance the NNI move. This was achieved by a bijection defined by Wakabayashi [15,Prop. 6.3].…”

Section: Weighted Nnis For {1 3}-graphsmentioning

confidence: 99%

“…Wakabayashi [15] gave a formula for two of the four constituent polynomials of the Ehrhart quasi-polynomial for P G and for the volume of P G when G is a connected cubic graph on n vertices. It is rather surprising that the Verlinde formula makes an appearance here, namely, Wakabayashi showed that, for odd t, L P G (t) = (t + 2) n/2 2 n+1 t+1 j=1 1 sin n πj t+2 and vol(P G ) = |B n | 2 n!…”

Section: Concluding Remarks and Open Problemsmentioning

confidence: 99%

See 2 more Smart Citations

Cubic Graphs, Their Ehrhart Quasi-Polynomials, and a Scissors Congruence Phenomenon

Fernandes

Pina

Alfonsín

et al. 2020

Discrete Comput Geom

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The scissors congruence conjecture for the unimodular group is an analogue of Hilbert's third problem, for the equidecomposability of polytopes. Liu and Osserman studied the Ehrhart quasi-polynomials of polytopes naturally associated to graphs whose vertices have degree one or three. In this paper, we prove the scissors congruence conjecture, posed by Haase and McAllister, for this class of polytopes.The key ingredient in the proofs is the nearest neighbor interchange on graphs and a naturally arising piecewise unimodular transformation. * FAPESP 13/03447-6 and 15/10323-7, CNPq 456792/2014-7 and 452507/2016-2, CAPES PROEX.

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Section: Lemma 7 the External Edges Of A Caterpillar Can Be Sorted Amentioning

confidence: 97%

“…To deal with weights on the edges of a {1, 3}-graph, we now enhance the NNI move. This was achieved by a bijection defined by Wakabayashi [15,Prop. 6.3].…”

Section: Weighted Nnis For {1 3}-graphsmentioning

confidence: 99%

Section: Concluding Remarks and Open Problemsmentioning

confidence: 99%

See 1 more Smart Citation

Cubic Graphs, Their Ehrhart Quasi-Polynomials, and a Scissors Congruence Phenomenon

Fernandes

Pina

Alfonsín

et al. 2020

Discrete Comput Geom

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“…They were mainly motivated by the relation of these quasi-polynomials to the study of dormant torally indigenous bundles on a general curve, objects arising in algebraic geometry [14]. This connection was further investigated in [17], and more properties of the polytope P G were presented in [6].…”

Section: Introductionmentioning

confidence: 99%

“…The polytope P G has nice geometric and combinatorial properties. For instance, Wakabayashi has proved [17,Proposition 5.3 and Corollary 5.4] that, for cubic graphs G 1 and G 2 , the polytopes P G 1 and P G 2 are isomorphic (that is, there is an R-linear bijection f : R d → R d such that P G 2 = f (P G 1 )) if and only if the graphs G 1 and G 2 are isomorphic. Let T be a {1, 3}-tree.…”

Section: Introductionmentioning

confidence: 99%

On the period collapse of a family of Ehrhart quasi-polynomials

Fernandes,

de Pina,

Alfonsín

et al. 2021

Preprint

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A graph whose nodes have degree 1 or 3 is called a {1, 3}-graph. Liu and Osserman associated a polytope to each {1, 3}-graph and studied the Ehrhart quasipolynomials of these polytopes. They showed that the vertices of these polytopes have coordinates in the set {0, 1 4 , 1 2 , 1}, which implies that the period of their Ehrhart quasi-polynomials is either 1, 2, or 4. We show that the period of the Ehrhart quasipolynomial of these polytopes is at most 2 if the graph is a tree or a cubic graph, and it is equal to 4 otherwise.In the process of proving this theorem, several interesting combinatorial and geometric properties of these polytopes were uncovered, arising from the structure of their associated graphs. The tools developed here may find other applications in the study of Ehrhart quasi-polynomials and enumeration problems for other polytopes that arise from graphs. Additionally, we have identified some interesting connections with triangulations of 3-manifolds.

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Cyclic étale coverings of generic curves and ordinariness of dormant opers

Wakabayashi

2023

Journal of Algebra

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Spin networks, Ehrhart quasipolynomials, and combinatorics of dormant indigenous bundles

Cited by 8 publications

References 16 publications

Cubic Graphs, Their Ehrhart Quasi-Polynomials, and a Scissors Congruence Phenomenon

Cubic Graphs, Their Ehrhart Quasi-Polynomials, and a Scissors Congruence Phenomenon

On the period collapse of a family of Ehrhart quasi-polynomials

Cyclic étale coverings of generic curves and ordinariness of dormant opers

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