December 2024
Algebraic & Geometric Topology
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December 2024
Algebraic & Geometric Topology
December 2024
For any pair of integers m and n such that , 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.
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.
October 2024
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2 Reads
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1 Citation
Algebraic & Geometric Topology
August 2024
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1 Read
Experimental Mathematics
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.
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 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 . 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.
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.
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) 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) of a knotted surface in 4-space. We further explore the combinatorics of pants decompositions to give sharp bounds for the L-invariant of large families of bridge trisections. As an application, we show that surfaces with L(S)≤2 must be unknotted.
July 2023
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1 Read
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.
... 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]. ...
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]. ...
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]. ...
September 2023
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. ...
Reference:
On keen weakly reducible bridge spheres
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). ...
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 . ...
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]: ...
August 2015