arXiv subjects

Cam Van Quach Hongler

Publications and source records attributed to Cam Van Quach Hongler.

7 recordsLinked to original sources

arXiv2022

Periodicity and Free Periodicity of Alternating Knots

A knot $K$ in $S^3$ is $q$-periodic if it admits a symmetry that is conjugate to a rotation of order $q$ of $S^3$. If $K$ admits a symmetry which is a homeomorphism without fixed point of period $q$ of $S^3$, then $K$ is called freely $q$-periodic. In a previous paper, we obtained, as a consequence of Flyping Theorem due to Menasco and Thislethwaite, that the $q$-periodicity with $q>2$ can be visualized in an alternating projection as a rotation of the projection sphere. In this paper, we show that the free $q$-action of an alternating knot can be represented on some alternating projection as a composition of a rotation of order $q$ with some flypes all occurring on the same twisted band diagram of its essential Conway decomposition. Therefore, for an alternating knot to be freely periodic, its essential decomposition must satisfy certain conditions. We show that any free or non-free $q$-action is somehow visible (virtually visible) and give some sufficient criteria to detect the existence of $q$-actions from virtually visible projections. Finally, we show how the Murasugi decomposition into atoms enables us to determine the visibility type $(q,r)$ of the freely $q$-periodic alternating knots ($(q,r)$-lens knots); in fact, we only need to focus on a certain atom of their Murasugi decomposition to deduce their visibility type.

math.GT↗

arXiv2021

On the Visibility of Alternating +Achiral Knots

This article is devoted to the study of prime alternating +achiral knots. In the case of arborescent knots, we prove in +AAA Visibility Theorem 5.1, that the symmetry is visible on a certain projection (not necessarily minimal) and that it is realised by a homeomorphism of order 4. In the general case (arborescent or not), if the prime alternating knot has no minimal projection on which +achirality is visible, we prove that the order of +achirality is necessarily equal to 4.

math.GT↗

arXiv2021

Periodic projections of alternating knots

This paper is devoted to prove the existence of $q$-periodic alternating projections of prime alternating $q$-periodic knots. The main tool is the Menasco-Thistlethwaite's Flyping theorem. Let $K$ be an oriented prime alternating knot that is $q$-periodic with $q\geq 3$, i.e. $K$ admits a symmetry that is a rotation of order $q$. Then $K$ has an alternating $q$-periodic projection. As applications, we obtain the crossing number of a $q$ -periodic alternating knot with $q\geq 3$ is a multiple of $q$ and we give an elementary proof that the knot $12_{a634} $ is not 3-periodic; this proof does not depend on computer computations as in "Periodic knots and Heegaard Floer correction terms" by Stanilav Jabuka and Swatee Naik (arXiv:1307.5116 [math.GT]).

math.GT↗

arXiv2011

On the Kawauchi conjecture about the Conway polynomial of achiral knots

We give a counterexample to the Kawauchi conjecture on the Conway polynomial of achiral knots which asserts that the Conway polynomial $C(z)$ of an achiral knot satisfies the splitting property $C(z)=F(z)F(-z)$ for a polynomial $F(z)$ with integer coefficients. We show that the Bonahon-Siebenmann decomposition of an achiral and alternating knot is reflected in the Conway polynomial. More explicitly, the Kawauchi conjecture is true for quasi-arborescent knots and counterexamples in the class of alternating knots must be quasi-polyhedral.

math.GT↗

arXiv2011

A proof of Tait's Conjecture on alternating-achiral knots

In this paper we are interested in symmetries of alternating knots, more precisely in those related to achirality. We call the following statement Tait's Conjecture on alternating -achiral knots: Let K be an alternating -achiral knot. Then there exists a minimal projection Π of K in S^2 \subset S^3 and an involution ϕ:S^3\toS^3 such that: 1) ϕ reverses the orientation of $S^3$; 2) ϕ(S^2) = S^2; 3) ϕ (Π) = Π; 4) ϕ has two fixed points on Π and hence reverses the orientation of K. The purpose of this paper is to prove this statement.

math.GT↗

arXiv2010

Prime order automorphisms of Klein surfaces representable by rotations of the euclidean space

Let S be a bordered orientable Klein surface and p a prime. Assume that f is an order p automorphism of S. In this work we obtain the conditions on the topological type of (S,f) to be conformally equivalent to (S',f') where S' is a bordered orientable Klein surface embedded in the Euclidean space and f' is the restriction to S' of a prime order rotation. We represent two famous automorphisms using rotations of R^4 and S^4 : the order seven automorphisms of the Klein quartic and the Wiman surface.

math.GT↗

arXiv2009

Link projections and flypes

Let Πbe a link projection in S^2. John Conway and later Francis Bonahon and Larry Siebenmann undertook to split $Π$ into canonical pieces. These pieces received different names: basic or polyhedral diagrams on one hand, rational, algebraic, bretzel, arborescent diagrams on the other hand. This paper proposes a thorough presentation of the theory, known to happy fews. We apply the existence and uniqueness theorem for the canonical decomposition to the classification of Haseman circles and to the localisation of the flypes.

math.GT↗