A Generalization of the Gram determinant of type A

The Gram determinant of type $A$ was introduced by Lickorish in his work on invariants of 3 - manifolds. We generalize the theory of the Gram determinant of type $A$ by evaluating, in the annulus, a bilinear form of non-intersecting connections in the disc. The main result provides a closed formula for this Gram determinant. We conclude the paper by discussing Chen's conjecture about the Gram determinant of type $Mb$ evaluated in the Klein bottle.Comment: 17 pages, 19 figure

On multiplying curves in the Kauffman bracket skein algebra of the thickened four-holed sphere

Based on the presentation of the Kauffman bracket skein algebra of the thickened torus given by the third author in previous work [4], Frohman and Gelca established a complete description of the multiplicative operation leading to a famous product-to-sum formula. In this paper, we study the multiplicative structure of the Kauffman bracket skein algebra of the thickened four-holed sphere. We present an algorithm to compute the product of any two elements of the algebra, and give an explicit formula for some families of curves. We surmise that the algorithm has quasi-polynomial growth with respect to the number of crossings of a pair of curves. Further, we conjecture the existence of a positive basis for the algebra

On framings of knots in 3-manifolds

We show that the only way of changing the framing of a knot or a link by ambient isotopy in an oriented $3$-manifold is when the manifold has a properly embedded non-separating $S^2$. This change of framing is given by the Dirac trick, also known as the light bulb trick. The main tool we use is based on McCullough's work on the mapping class groups of $3$-manifolds. We also relate our results to the theory of skein modules.Comment: 12 pages, 9 figure

A Counterexample to the Generalisation of Witten's Conjecture

This note provides a counterexample to a conjecture by Marché about the structure of the Kauffman bracket skein module for closed compact oriented 3-manifolds over the ring of Laurent polynomials.ISSN:0271-4132ISSN:1098-362

The Generalized Kauffman-Harary Conjecture is True

For a reduced alternating diagram of a knot with a prime determinant �, the Kau man-Harary conjecture states that every non-trivial Fox �-coloring of the knot assigns di erent colors to its arcs. In this paper, we prove a generalization of the conjecture stated nineteen years ago by Asaeda, Przytycki, and Sikora: for every pair of distinct arcs in the reduced alternating diagram of a prime link with determinant �, there exists a Fox �-coloring that distinguishes them

Effect of Rhododendron arboreum Leaf Extract on the Antioxidant Defense System against Chromium (VI) Stress in Vigna radiata Plants

In the current investigation, we studied role of Rhododendron leaf extract in Vigna radiata grown under chromium metal stress. We observed that seed treatment with Rhododendron leaf extract resulted in the recuperation of seedling growth under chromium toxicity. Seed treatment with Rhododendron leaf extract significantly improved the contents of anthocyanin and xanthophyll pigments under stress. The antioxidative defense system triggered after Rhododendron extract treatment, resulting in the increased actions of antioxidant enzymes. Oxidative stress induced by the assembly of reactive oxygen species was reduced after Rhododendron extract treatment under chromium toxicity as indicated by the enhanced contents of non-enzymatic antioxidants, namely ascorbic acid, tocopherol, and glutathione. Furthermore, Rhododendron leaf extract treatment under chromium metal stress also encouraged the biosynthesis of organic acids, polyphenols, as well as amino acids in Vigna radiata. Statistical analysis of the data with multiple linear regression also supported that Rhododendron leaf extract can effectively ease chromium metal-induced phytotoxicity in Vigna radiata

Ecological and human health risks appraisal of metal(loid)s in agricultural soils: a review

Agriculture is one of the major human activities that changed the landforms, water resources and the biogeochemical cycles. Pollution of agricultural soilsby metal(loid)s is a serious and global hazard but worldwide studies related to metal(loid)s pollution in agricultural soils are very limited. To fulfil this gap, metal(loid)s content in agricultural soils from 2001 to 2019 all over the world was reviewed. Multivariate statistical techniques, contamination indices and human health risk assessment were determined for the metal(loid)s. Among the analysed metal(loid)s, the average contents of Zn, Cu, Pb, Cr, Cd, As and Ni exceeded the Canadian, and China soil guidelines limits. The results of contamination factor indicated that Cr, Pb, Cd, As and Zn are the key pollution contaminants. As and Cd had the highest enrichment among the analysed metal(loid)s according to the enrichment factor. The potential and modified ecological risk index showed that Cd is the foremost contaminant responsible for ecological threats. The non-carcinogenic risk for ingestion pathway indicated that As, Pb and Cr are the foremost contaminants responsible for affecting human health, while dermal pathway results showed less risk of metal(loid)s in the agricultural soils. The carcinogenic risk revealed that As, Pb and Cr are the key contaminants that affects human health

Amelioration of Chlorpyrifos-Induced Toxicity in Brassica juncea L. by Combination of 24-Epibrassinolide and Plant-Growth-Promoting Rhizobacteria

Pervasive use of chlorpyrifos (CP), an organophosphorus pesticide, has been proven to be fatal for plant growth, especially at higher concentrations. CP poisoning leads to growth inhibition, chlorosis, browning of roots and lipid and protein degradation, along with membrane dysfunction and nuclear damage. Plants form a linking bridge between the underground and above-ground communities to escape from the unfavourable conditions. Association with beneficial rhizobacteria promotes the growth and development of the plants. Plant hormones are crucial regulators of basically every aspect of plant development. The growing significance of plant hormones in mediating plant–microbe interactions in stress recovery in plants has been extensively highlighted. Hence, the goal of the current study was to investigate the effect of 24-epibrassinolide (EBL) and PGPRs (Pseudomonas aeruginosa (Ma), Burkholderia gladioli (Mb)) on growth and the antioxidative defence system of CP-stressed Brassica juncea L. seedlings. CP toxicity reduced the germination potential, hypocotyl and radicle development and vigour index, which was maximally recuperated after priming with EBL and Mb. CP-exposed seedlings showed higher levels of superoxide anion (O2−), hydrogen peroxide (H2O2), lipid peroxidation and electrolyte leakage (EL) and a lower level of nitric oxide (NO). In-vivo visualisation of CP-stressed seedlings using a light and fluorescent microscope also revealed the increase in O2−, H2O2 and lipid peroxidation, and decreased NO levels. The combination of EBL and PGPRs reduced the reactive oxygen species (ROS) and malondialdehyde (MDA) contents and improved the NO level. In CP-stressed seedlings, increased gene expression of defence enzymes such as superoxide dismutase (SOD), ascorbate peroxidase (APOX), glutathione peroxidase (GPOX), dehydroascorbate reductase (DHAR) and glutathione reductase (GPOX) was seen, with the exception of catalase (CAT) on supplementation with EBL and PGPRs. The activity of nitrate reductase (NR) was likewise shown to increase after treatment with EBL and PGPRs. The results obtained from the present study substantiate sufficient evidence regarding the positive association of EBL and PGPRs in amelioration of CP-induced oxidative stress in Brassica juncea seedlings by strengthening the antioxidative defence machinery