Nicholas Cazet’s research while affiliated with University of California, Davis and other places

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Publications (15)


Quandles with one nontrivial column
  • Article

February 2024

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3 Reads

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1 Citation

Journal of Algebra and Its Applications

Nicholas Cazet

The axioms of a quandle imply that the columns of its Cayley table are permutations. This paper studies quandles with exactly one non-trivially permuted column. Their automorphism groups, quandle polynomials, (symmetric) cohomology groups, and [Formula: see text] quandles are studied. The quiver and cocycle invariant of links using these quandles are shown to relate to linking number.


Bounds in simple hexagonal lattice and Classification of 11-stick knots

January 2024

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8 Reads

Journal of Knot Theory and Its Ramifications

Yueheng Bao

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Ari Benveniste

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Marion Campisi

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[...]

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Ethan Sherman

The stick number and the edge length of a knot type in the simple hexagonal lattice (sh-lattice) are the minimal numbers of sticks and edges required, respectively, to construct a knot of the given type in sh-lattice. By introducing a linear transformation between lattices, we prove that for any given knot both values in the sh-lattice are strictly less than the values in the cubic lattice. Finally, we show that the only non-trivial [Formula: see text]-stick knots in the sh-lattice are the trefoil knot ([Formula: see text]) and the figure-eight knot ([Formula: see text]).


On the triple point number of surface-links in Yoshikawa’s table

May 2023

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2 Reads

Journal of Knot Theory and Its Ramifications

Yoshikawa made a table of knotted surfaces in [Formula: see text] with ch-index 10 or less. This remarkable table is the first to enumerate knotted surfaces analogous to the classical prime knot table. A broken sheet diagram of a surface-link is a generic projection of the surface in [Formula: see text] with crossing information along its singular set. The minimal number of triple points among all broken sheet diagrams representing a given surface-link is its triple point number. This paper compiles the known triple point numbers of the surface-links represented in Yoshikawa’s table and calculates or provides bounds on the triple point number of the remaining surface-links.


The stick number of rail arcs

April 2023

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19 Reads

Journal of Knot Theory and Its Ramifications

Consider two parallel lines [Formula: see text] and [Formula: see text] in [Formula: see text]. A rail arc is an embedding of an arc in [Formula: see text] such that one endpoint is on [Formula: see text], the other is on [Formula: see text], and its interior is disjoint from [Formula: see text]. Rail arcs are considered up to rail isotopies, ambient isotopies of [Formula: see text] with each self-homeomorphism mapping [Formula: see text] and [Formula: see text] onto themselves. When the manifolds and maps are taken in the piecewise linear category, these rail arcs are called stick rail arcs. The stick number of a rail arc class is the minimum number of sticks, line segments in a p.l. arc, needed to create a representative. This paper calculates the stick number of rail arcs classes with a crossing number at most 2 and uses a winding number invariant to calculate the stick numbers of infinitely many rail arc classes. Each rail arc class has two canonically associated knot classes, its under and over companions. This paper also introduces the rail stick number of knot classes, the minimum number of sticks needed to create a rail arcs whose under or over companion is the knot class. The rail stick number is calculated for 29 knot classes with crossing number at most 9. The stick number of multi-component rail arcs classes is considered as well as the lattice stick number of rail arcs.


Quandles with one non-trivial column

March 2023

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12 Reads

The axioms of a quandle imply that the columns of its Cayley table are permutations. This paper studies quandles with exactly one non-trivially permuted column. Their automorphism groups, quandle polynomials, (symmetric) cohomology groups, and Hom quandles are studied. The quiver and cocycle invariant of links using these quandles are shown to relate to linking number.


Stable irreducibility via the symmetric quandle cocycle invariant

December 2022

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2 Reads

A new variation of the symmetric quandle cocycle invariant of surface-knots is introduced and used to prove the existence of stably irreducible, non-orientable surface-knots in S4S^4 of arbitrary even genus. Additionally, the symmetric quandle 3-cocycle invariant is shown to obstruct ribbon concordance between non-orientable surface-links, namely the 2-component links of projective planes from Yoshikawa's table.


Bounds in simple hexagonal lattice and classification of 11-stick knots

November 2022

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15 Reads

The stick number and the edge length of a knot type in the simple hexagonal lattice (sh-lattice) are the minimal numbers of sticks and edges required, respectively, to construct a knot of the given type in sh-lattice. By introducing a linear transformation between lattices, we prove that for any given knot both values in the sh-lattice are strictly less than the values in the cubic lattice. Moreover, we find lower bounds for any given knot's stick number and edge length in sh-lattice using these properties in the cubic lattice. Finally, we show that the only non-trivial 11-stick knots in the sh-lattice are the trefoil knot (313_1) and the figure-eight knot (414_1).


Vertex distortion detects the unknot

October 2022

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6 Reads

Journal of Knot Theory and Its Ramifications

The first two authors introduced vertex distortion of lattice knots and showed that the vertex distortion of the unknot is 1. It was conjectured that the vertex distortion of a knot class is 1 if and only if it is trivial. We use Denne and Sullivan’s lower bound on Gromov distortion to bound the vertex distortion of non-trivial lattice knots. This bounding allows us to conclude that a knot class has vertex distortion 1 if and only if it is trivial. We also show that vertex distortion does not have a universal upper bound and provide a vertex distortion calculator.


Surface-link families with arbitrarily large triple point number

August 2022

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4 Reads

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2 Citations

Topology and its Applications

Analogous to a classical link diagram, a surface-link can be generically projected to 3-space and given crossing information to create a broken sheet diagram. The triple point number of a surface-link is the minimal number of triple points among all broken sheet diagrams that lift to that surface-link. This paper generalizes a family of Oshiro to show that there are non-split surface-links of arbitrarily many trivial components whose triple point number can be made arbitrarily large.


The stick number of rail arcs

June 2022

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18 Reads

Consider two parallel lines 1\ell_1 and 2\ell_2 in R3\mathbb{R}^3. A rail arc is an embedding of an arc in R3\mathbb{R}^3 such that one endpoint is on 1\ell_1, the other is on 2\ell_2, and its interior is disjoint from 12\ell_1\cup\ell_2. Rail arcs are considered up to rail isotopies, ambient isotopies of R3\mathbb{R}^3 with each self-homeomorphism mapping 1\ell_1 and 2\ell_2 onto themselves. When the manifolds and maps are taken in the piecewise linear category, these rail arcs are called stick rail arcs. The stick number of a rail arc class is the minimum number of sticks, line segments in a p.l. arc, needed to create a representative. This paper will calculate the stick numbers of rail arcs classes with a crossing number at most 2 and use a winding number invariant to calculate the stick numbers of infinitely many rail arc classes. Each rail arc class has two canonically associated knot classes, its under and over companions. This paper also introduces the rail stick number of knot classes, the minimum number of sticks needed to create a rail arcs whose under or over companion is the knot class. The rail stick number is calculated for all knot classes with crossing number at most 9. The stick number of multi-component rail arcs classes is considered as well as the lattice stick number of rail arcs.


Citations (3)


... For some particular choices of target quandles X appearing in Hom(Q(L), X ), the quandle coloring quivers have been determined for various families of links [1,4,16]. It has also been shown that in some cases, the quiver gives more information than cocycle and module enhancements [5,9]. ...

Reference:

Quandle coloring quivers and 2-bridge links
Quandles with one nontrivial column
  • Citing Article
  • February 2024

Journal of Algebra and Its Applications