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Register a new userCounterexamples to the quadrisecant approximation conjecture
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A quadrisecant of a knot is a straight line intersecting the knot at four points. If a knot has finitely many quadrisecants, one can replace each subarc between two adjacent secant points by the line segment between them to get the quadrisecant approximation of the original knot. It was conjectured that the quadrisecant approximation is always a knot with the same knot type as the original knot. We show that every knot type contains two knots, the quadrisecant approximation of one knot has self intersections while the quadrisecant approximation of the other knot is a knot with different knot type.
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Hedetniemi conjectured in 1966 that $chi(G times H) = min{chi(G), chi(H)}$ for all graphs $G$ and $H$. Here $Gtimes H$ is the graph with vertex set $ V(G)times V(H)$ defined by putting $(x,y)$ and $(x,y)$ adjacent if and only if $xxin E(G)$ and $yyin E(H)$. This conjecture received a lot of attention in the past half century. Recently, Shitov refuted this conjecture. Let $p$ be the minimum number of vertices in a graph of odd girth $7$ and fractional chromatic number greater than $3+4/(p-1)$. Shitovs proof shows that Hedetniemis conjecture fails for some graphs with chromatic number about $p^22^{p+1} $ and with about $(p^22^{p+1})^{p^32^{p-1}} $ vertices. In this paper, we show that the conjecture fails already for some graphs $G$ and $H$ with chromatic number $3lceil frac {p+1}2 rceil $ and with $p lceil (p-1)/2 rceil$ and $3 lceil frac {p+1}2 rceil (p+1)-p$ vertices, respectively. The currently known upper bound for $p$ is $148$. Thus Hedetniemis conjecture fails for some graphs $G$ and $H$ with chromatic number $225$, and with $10,952$ and $33,377$ vertices, respectively.
We give improved separations for the query complexity analogue of the log-approximate-rank conjecture i.e. we show that there are a plethora of total Boolean functions on $n$ input bits, each of which has approximate Fourier sparsity at most $O(n^3)$ and randomized parity decision tree complexity $Theta(n)$. This improves upon the recent work of Chattopadhyay, Mande and Sherif (JACM 20) both qualitatively (in terms of designing a large number of examples) and quantitatively (improving the gap from quartic to cubic). We leave open the problem of proving a randomized communication complexity lower bound for XOR compositions of our examples. A linear lower bound would lead to new and improved refutations of the log-approximate-rank conjecture. Moreover, if any of these compositions had even a sub-linear cost randomized communication protocol, it would demonstrate that randomized parity decision tree complexity does not lift to randomized communication complexity in general (with the XOR gadget).
The Hall ratio of a graph $G$ is the maximum value of $v(H) / alpha(H)$ taken over all non-null subgraphs $H$ of $G$. For any graph, the Hall ratio is a lower-bound on its fractional chromatic number. In this note, we present various constructions of graphs whose fractional chromatic number grows much faster than their Hall ratio. This refutes a conjecture of Harris.
In this short note we report on results on a computational search for a counterexample to the strong coincidence conjecture. In particular, we discuss the method used so that further searches can be conducted.
The Hilbert-Smith Conjecture states that if G is a locally compact group which acts effectively on a connected manifold as a topological transformation group, then G is a Lie group. A rather straightforward proof of this conjecture is given. The motivation is work of Cernavskii (``Finite-to-one mappings of manifolds, Trans. of Math. Sk. 65 (107), 1964.) His work is generalized to the orbit map of an effective action of a p-adic group on compact connected n-manifolds with the aid of some new ideas. There is no attempt to use Smith Theory even though there may be similarities. It is well known that if a locally compact group acts effectively on a connected n-manifold M and G is not a Lie group, then there is a subgroup H of G isomorphic to a p-adic group A_p which acts effectively on M. It can be shown that A_p can not act effectively on an n-manifold and, hence, The Hilbert Smith Conjecture is true. The existence of a non empty fixed point set adds some complexity to the proof. In this paper, it is shown that A_p can not act freely on a compact connected n-manifold. The basic ideas for the general case are more clearly seen in this case. The general proof will be given in another paper.
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