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Register a new userA lower bound for the number of Reidemeister moves of type III
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J. Scott Carter
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We study the number of Reidemeister type III moves using Fox n-colorings of knot diagrams.
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We prove that any diagram of the unknot with c crossings may be reduced to the trivial diagram using at most (236 c)^{11} Reidemeister moves. Moreover, every diagram in this sequence has at most (7 c)^2 crossings. We also prove a similar theorem for split links, which provides a polynomial upper bound on the number of Reidemeister moves required to transform a diagram of the link into a disconnected diagram.
The dual to a tetrahedron consists of a single vertex at which four edges and six faces are incident. Along each edge, three faces converge. A 2-foam is a compact topological space such that each point has a neighborhood homeomorphic to a neighborhood of that complex. Knotted foams in 4-dimensional space are to knotted surfaces, as knotted trivalent graphs are to classical knots. The diagram of a knotted foam consists of a generic projection into 4-space with crossing information indicated via a broken surface. In this paper, a finite set of moves to foams are presented that are analogous to the Reidemeister-type moves for knotted graphs. These moves include the Roseman moves for knotted surfaces. Given a pair of diagrams of isotopic knotted foams there is a finite sequence of moves taken from this set that, when applied to one diagram sequentially, produces the other diagram.
We approximate intersection numbers $biglangle psi_1^{d_1}cdots psi_n^{d_n}bigrangle_{g,n}$ on Deligne-Mumfords moduli space $overline{mathcal M}_{g,n}$ of genus $g$ stable complex curves with $n$ marked points by certain closed-form expressions in $d_1,dots,d_n$. Conjecturally, these approximations become asymptotically exact uniformly in $d_i$ when $gtoinfty$ and $n$ remains bounded or grows slowly. In this note we prove a lower bound for the intersection numbers in terms of the above-mentioned approximatingexpressions multiplied by an explicit factor $lambda(g,n)$, which tends to $1$ when $gtoinfty$ and $d_1+dots+d_{n-2}=o(g)$.
In [13] the authors show that if $mu$ is a strongly compact cardinal, $K$ is an Abstract Elementary Class (AEC) with $LS(K)<mu$, and $K$ satisfies joint embedding (amalgamation) cofinally below $mu$, then $K$ satisfies joint embedding (amalgamation) in all cardinals $ge mu$. The question was raised if the strongly compact upper bound was optimal. In this paper we prove the existence of an AEC $K$ that can be axiomatized by an $mathcal{L}_{omega_1,omega}$-sentence in a countable vocabulary, so that if $mu$ is the first measurable cardinal, then (1) $K$ satisfies joint embedding cofinally below $mu$ ; (2) $K$ fails joint embedding cofinally below $mu$; and (3) $K$ satisfies joint embedding above $mu$. Moreover, the example can be generalized to an AEC $K^chi$ axiomatized in $mathcal{L}_{chi^+, omega}$, in a vocabulary of size $chi$, such that (1)-(3) hold with $mu$ being the first measurable above $chi$. This proves that the Hanf number for joint embedding is contained in the interval between the first measurable and the first strongly compact. Since these two cardinals can consistently coincide, the upper bound from [13] is consistently optimal. This is also the first example of a sentence whose joint embedding spectrum is (consistently) neither an initial nor an eventual interval of cardinals. By Theorem 3.26, it is consistent that for any club $C$ on the first measurable $mu$, JEP holds exactly on $lim C$ and everywhere above $mu$.
Every link in the 3-sphere has a projection to the plane where the only singularities are pairwise transverse triple points. The associated diagram, with height information at each triple point, is a triple-crossing diagram of the link. We give a set of diagrammatic moves on triple-crossing diagrams analogous to the Reidemeister moves on ordinary diagrams. The existence of n-crossing diagrams for every n>1 allows the definition of the n-crossing number. We prove that for any nontrivial, nonsplit link, other than the Hopf link, its triple-crossing number is strictly greater than its quintuple-crossing number.
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