Khovanov Homology & Uniqueness of Surfaces in the 4-ball

We use the functoriality of Khovanov homology to examine the smooth, boundary-preserving isotopy of surfaces embedded in the 4-ball. We exemplify an infinite family of prime knots that bound an arbitrarily-large number of smoothly-distinct slice disks by distinguishing the maps they induce on Khovanov homology. Similar techniques produce an infinite family of knots that each bound a pair of exotic surfaces of arbitrary genus

Relative Khovanov-Jacobsson classes

To a smooth, compact, oriented, properly-embedded surface in the $4$-ball, we define an invariant of its boundary-preserving isotopy class from the Khovanov homology of its boundary link. Previous work showed that when the boundary link is empty, this invariant is determined by the genus of the surface. We show that this relative invariant: can obstruct sliceness of knots; detects a pair of slices for $9_{46}$; is not hindered by detecting connected sums with knotted $2$-spheres.Comment: 16 pages, 11 figure

Seifert surfaces in the 4-ball

We answer a question of Livingston from 1982 by producing Seifert surfaces of the same genus for a knot in $S^3$ that do not become isotopic when their interiors are pushed into $B^4$. In particular, we identify examples where the surfaces are not even topologically isotopic in $B^4$, examples that are topologically but not smoothly isotopic, and examples of infinite families of surfaces that are distinct only up to isotopy rel. boundary. Our main proofs distinguish surfaces using the cobordism maps on Khovanov homology, and our calculations demonstrate the stability and computability of these maps under certain satellite operations.Comment: 31 pages + bibliography, 28 figures. Some computational details available in ancillary file. Compared to v1, we added Theorems 1.4 and 1.5 producing infinite families of Seifert surfaces that are pairwise not isotopic rel. boundary in B^4. (In v3, just corrected floats in Fig. 27.

SNP-SNP interactions in breast cancer susceptibility

BACKGROUND: Breast cancer predisposition genes identified to date (e.g., BRCA1 and BRCA2) are responsible for less than 5% of all breast cancer cases. Many studies have shown that the cancer risks associated with individual commonly occurring single nucleotide polymorphisms (SNPs) are incremental. However, polygenic models suggest that multiple commonly occurring low to modestly penetrant SNPs of cancer related genes might have a greater effect on a disease when considered in combination. METHODS: In an attempt to identify the breast cancer risk conferred by SNP interactions, we have studied 19 SNPs from genes involved in major cancer related pathways. All SNPs were genotyped by TaqMan 5'nuclease assay. The association between the case-control status and each individual SNP, measured by the odds ratio and its corresponding 95% confidence interval, was estimated using unconditional logistic regression models. At the second stage, two-way interactions were investigated using multivariate logistic models. The robustness of the interactions, which were observed among SNPs with stronger functional evidence, was assessed using a bootstrap approach, and correction for multiple testing based on the false discovery rate (FDR) principle. RESULTS: None of these SNPs contributed to breast cancer risk individually. However, we have demonstrated evidence for gene-gene (SNP-SNP) interaction among these SNPs, which were associated with increased breast cancer risk. Our study suggests cross talk between the SNPs of the DNA repair and immune system (XPD-[Lys751Gln] and IL10-[G(-1082)A]), cell cycle and estrogen metabolism (CCND1-[Pro241Pro] and COMT-[Met108/158Val]), cell cycle and DNA repair (BARD1-[Pro24Ser] and XPD-[Lys751Gln]), and within carcinogen metabolism (GSTP1-[Ile105Val] and COMT-[Met108/158Val]) pathways. CONCLUSION: The importance of these pathways and their communication in breast cancer predisposition has been emphasized previously, but their biological interactions through SNPs have not been described. The strategy used here has the potential to identify complex biological links among breast cancer genes and processes. This will provide novel biological information, which will ultimately improve breast cancer risk management