Puttipong Pongtanapaisan’s research while affiliated with Arizona State University and other places

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Publications (24)


On keen weakly reducible bridge spheres
  • Article

December 2024

Algebraic & Geometric Topology

Puttipong Pongtanapaisan

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Daniel Rodman

Bridge Positions of Links That Cannot Be Monotonically Simplified

December 2024

For any pair of integers m and n such that 3<m<n3<m<n, we provide an infinite family of links, where each link in the family has a locally minimal n-bridge position and a globally minimal m-bridge position. We accomplish this by applying the criterion of Takao et al. The n-bridge position is interesting because the corresponding bridge sphere is unperturbed, so it must be perturbed at least once before it can be de-perturbed to attain a globally minimal m-bridge sphere.


Bridge indices of spatial graphs and diagram colorings

October 2024

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5 Reads

We extend the Wirtinger number of links, an invariant originally defined by Blair et al. in terms of extending initial colorings of some strands of a diagram to the entire diagram, to spatial graphs. We prove that the Wirtinger number equals the bridge index of spatial graphs, and we implement an algorithm in Python which gives a more efficient way to estimate upper bounds of bridge indices. Combined with lower bounds from diagram colorings by elements from certain algebraic structures, we obtain exact bridge indices for a large family of spatial graphs.




The relation for the fundamental quandle at each crossing
Left: The [0] tangle. Middle: The [+2]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[+2]$$\end{document} tangle. Right: The [-4]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[-4]$$\end{document} tangle
Construction of T1T2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_1T_2$$\end{document} from T1,T2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_1,T_2$$\end{document}
Labels for the strands of a 2-bridge link
The figure-eight knot viewed as the two-bridge knot

+3

Quandle coloring quivers and 2-bridge links
  • Article
  • Publisher preview available

June 2024

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27 Reads

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1 Citation

European Journal of Mathematics

The quandle coloring quiver was introduced by Cho and Nelson as a categorification of the quandle coloring number. In some cases, it has been shown that the quiver invariant offers more information than other quandle enhancements. In this paper, we compute the quandle coloring quivers of 2-bridge links with respect to the dihedral quandles.

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Bridge number and meridional rank of knotted surfaces

December 2023

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3 Reads

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1 Citation

Canadian Journal of Mathematics

The meridional rank conjecture asks whether the bridge number of a knot in S3S^3 is equal to the minimal number of meridians needed to generate the fundamental group of its complement. In this paper, we investigate the analogous conjecture for knotted spheres in S4S^4 . Towards this end, we give a construction to produce classical knots with quotients sending meridians to elements of any finite order in Coxeter groups and alternating groups, which detect their meridional ranks. We establish the equality of bridge number and meridional rank for these knots and knotted spheres obtained from them by twist-spinning. On the other hand, we show that the meridional rank of knotted spheres is not additive under connected sum, so that either bridge number also collapses, or meridional rank is not equal to bridge number for knotted spheres.


Spatial graphoids

September 2023

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137 Reads

Aequationes mathematicae

To study knotted graphs with open ends arising in proteins, we introduce 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. By analyzing the Yamada polynomial, we provide bounds for the crossing numbers. As an application, we can produce nontrivial graphoids by verifying that they satisfy adequacy conditions in the same spirit as Lickorish and Thistlethwaite’s notion of adequate links.


Bounds for Kirby–Thompson invariants of knotted surfaces

September 2023

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12 Reads

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2 Citations

Geometriae Dedicata

Blair, Campisi, Taylor, and Tomova introduced a non-negative integer-valued invariant L(S)L(S){\mathcal {L}}(S) of a smooth surface S in the 4-sphere. In this paper, we extend previous work done by the authors with Scott Taylor to compute the invariant L(S)L(S){\mathcal {L}}(S) of a knotted surface in 4-space. We further explore the combinatorics of pants decompositions to give sharp bounds for the LL{\mathcal {L}}-invariant of large families of bridge trisections. As an application, we show that surfaces with L(S)≤2L(S)2{\mathcal {L}}(S)\le 2 must be unknotted.


Pants distances of knotted surfaces in 4-manifolds

July 2023

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1 Read

Román Aranda

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Sarah Blackwell

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Devashi Gulati

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[...]

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Puttipong Pongtanapaisan

We define a pants distance for knotted surfaces in 4-manifolds which generalizes the complexity studied by Blair-Campisi-Taylor-Tomova for surfaces in the 4-sphere. We determine that if the distance computed on a given diagram does not surpass a theoretical bound in terms of the multisection genus, then the (4-manifold, surface) pair has a simple topology. Furthermore, we calculate the exact values of our invariants for many new examples such as the spun lens spaces. We provide a characterization of genus two quadrisections with distance at most six.


Citations (7)


... This invariant was extended to the case where X is a 4-manifold with boundary in [4]. It was also adapted to knotted surfaces using bridge trisections in [3], see also [1,2]. ...

Reference:

Some lower bounds for the Kirby-Thompson invariant
Bounding the Kirby–Thompson invariant of spun knots
  • Citing Article
  • October 2024

Algebraic & Geometric Topology

... Any surface can be described using a triplane diagram. Since their introduction, triplane diagrams have been used to study known surface-link invariants such as colorings [ST22], Seifert solids [JMMZ22b], triple point numbers [JMMZ22a], and the fundamental group of the complement [JP23]. Examples of new invariants obtained from triplane diagrams are the crossing number [AAD + 23], group trisections [BKK + 24], Nielsen equivalence [JMMZ22a], and L-invariants [BCTT22,APZ23]. ...

Bridge number and meridional rank of knotted surfaces
  • Citing Article
  • December 2023

Canadian Journal of Mathematics

... Since their introduction, triplane diagrams have been used to study known surface-link invariants such as colorings [ST22], Seifert solids [JMMZ22b], triple point numbers [JMMZ22a], and the fundamental group of the complement [JP23]. Examples of new invariants obtained from triplane diagrams are the crossing number [AAD + 23], group trisections [BKK + 24], Nielsen equivalence [JMMZ22a], and L-invariants [BCTT22,APZ23]. ...

Bounds for Kirby–Thompson invariants of knotted surfaces

Geometriae Dedicata

... One can ask the interesting question of which surfaces are topologically minimal. Many authors have given examples of topologically minimal Heegaard surfaces [3,15,5,6] and bridge surfaces [16,25,22]. Heegaard surfaces that are not topologically minimal have also been studied by several authors who constructed keen weakly reducible Heegaard surfaces. ...

Critical bridge spheres for links with arbitrarily many bridges
  • Citing Article
  • August 2020

Revista Matemática Complutense

... crossing changes, Δ-moves, 4-moves, forbidden moves), we can find a fixed polynomial P such that V L1 − V L2 is always a multiple of P . Additional results of this kind were studied in [11] (C n -moves) and [1] (double-Δ-moves). ...

Oriented Local Moves and Divisibility of the Jones-Kauffman Polynomial
  • Citing Article
  • November 2019

Journal of Knot Theory and Its Ramifications

... Further, our algorithm can be adapted to accept non-planar Gauss codes and thus to give lower bounds on the virtual bridge number of virtual knots. When paired with the upper bounds from [23], this technique can be used compute the virtual bridge number of many virtual knots. Finally, our algorithm can be used in conjunction with the results in [16] to establish the meridional rank and bridge number of certain twist spun knots, which are knotted 2-spheres in R 4 . ...

Wirtinger Numbers for Virtual Links
  • Citing Article
  • January 2018

Journal of Knot Theory and Its Ramifications

... Then, we use Burnside's lemma to calculate r 8 2 Idea of calculating r n using Burnside's lemma Burnside's lemma is a standard combinatorial tool for counting the orbits of set under group action. Let G denote a finite group that acts upon a set X. Burnside's lemma asserts that the number of orbits |X/G| with respect to the action equals the average size of the sets X g = {x ∈ X | gx = x} when ranging over each g ∈ G [6,7]: ...

Annular Non-Crossing Matchings
  • Citing Article
  • August 2015