In 1996, Cromwell and Nutt [7] found an upper bound on the arc index which
is related to the minimal crossing number and conjectured that the upper bound
achieves the lowest possible index for alternating links.CONJECTURE. Let L be any prime link. Then α(L) [les ] c(L)+2.
Moreover this inequality is strict if and only if L is not alternating.In this paper, we define a new diagram, called a wheel diagram, of a link and use
it to prove this conjecture.
We adapt Thistlethwaite's alternating tangle decomposition of a knot diagram to identify the potential extreme terms in its bracket polynomial, and give a simple combinatorial calculation for their coefficients, based on the intersection graph of certain chord diagrams.
To gain insight into early embryo development, we utilized microarray technology to compare gene expression profiles in four-cell (4C), morula (MO), and blastocyst (BL) stage embryos. Differences in spot intensities were normalized, and grouped by using Avadis Prophetic software platform (version 3.3, Strand Genomics Ltd.) and categories were based on the PANTHER and gene ontology (GO) classification system. This technique identified 622 of 7,927 genes as being more highly expressed in MO when compared to 4C (P < 0.05); similarly, we identified 654 of 9,299 genes as being more highly expressed in BL than in MO (P < 0.05). Upregulation of genes for cytoskeletal, cell adhesion, and cell junction proteins were identified in the MO as compared to the 4C stage embryos, this means they could be involved in the cell compaction necessary for the development to the MO. Genes thought to be involved in ion channels, membrane traffic, transfer/carrier proteins, and lipid metabolism were also identified as being expressed at a higher level in the BL stage embryos than in the MO. Real-time RT-PCR was performed to confirm differential expression of selected genes. The identification of the genes being expressed in here will provide insight into the complex gene regulatory networks effecting compaction and blastocoel formation.
In this paper, we study the colorability of link diagrams by the Alexander quandles. We show that if the reduced Alexander polynomial ∆ L (t) is vanishing, then L admits a nontrivial coloring by any non-trivial Alexander quandle Q, and that if ∆ L (t) = 1, then L admits only the trivial coloring by any Alexander quandle Q, also show that if ∆ L (t) = 0, 1, then L admits a non-trivial coloring by the Alexander quandle Λ/(∆ L (t)).
In this paper, we study the Gauss diagrams for periodic virtual knots (Theorem 3.1) and show that the virtual knot corresponding to a periodic Gauss diagram is equivalent to the periodic virtual knot whose factor is the virtual knot corresponding to the factor Gauss diagram (Theorem 3.2). We give formulae for the writhe polynomial and the affine index polynomial of periodic virtual knots by using those of factor knots (Corollary 4.2, Corollary 4.6).
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