Neighbors of knots in the Gordian graph

We show that every knot is one crossing change away from a knot of arbitrarily high bridge number and arbitrarily high bridge distance.Comment: Accepted by American Mathematical Monthly. New version incorporates referee comment

The relative L\mathcal{L}-invariant of a compact 44-manifold

In this paper, we introduce the relative $\mathcal{L}$-invariant $r\mathcal{L}(X)$ of a smooth, orientable, compact 4-manifold $X$ with boundary. This invariant is defined by measuring the lengths of certain paths in the cut complex of a trisection surface for $X$. This is motivated by the definition of the $\mathcal{L}$-invariant for smooth, orientable, closed 4-manifolds by Kirby and Thompson. We show that if $X$ is a rational homology ball, then $r\mathcal{L}(X)=0$ if and only if $X\cong B^4$. In order to better understand relative trisections, we also produce an algorithm to glue two relatively trisected 4-manifold by any Murasugi sum or plumbing in the boundary, and also prove that any two relative trisections of a given 4-manifold $X$ are related by interior stabilization, relative stabilization, and the relative double twist, which we introduce in this paper as a trisection version of one of Piergallini and Zuddas's moves on open book decompositions. Previously, it was only known (by Gay and Kirby) that relative trisections inducing equivalent open books on $X$ are related by interior stabilizations.Comment: 39 pages, 14 figures. v2: Significantly improved discussion of relative double twist (Section 2), changed some statements in Section 4, restructured Sections 3 and

What Powers the 2006 Outburst of the Symbiotic Star BF Cygni?

BF Cygni is a classical symbiotic binary. Its optical light curve occasionally shows outbursts of the Z And-type, whose nature is not well understood. During the 2006 August, BF Cyg underwent the recent outburst, and continues its active phase to the present. The aim of this contribution is to determine the fundamental parameters of the hot component in the binary during the active phase. For this purpose we used a high- and low-resolution optical spectroscopy and the multicolour UBV RCIC photometry. Our photometric monitoring revealed that a high level of the star’s brightness lasts for unusually long time of > 7 years. A sharp violet-shifted absorption component and broad emission wings in the Hα profile developed during the whole active phase. From 2009, our spectra revealed a bipolar ejection from the white dwarf (WD). Modelling the spectral energy distribution (SED) of the low-resolution spectra showed simultaneous presence of a warm (< 10 000 K) disk-like pseudophotosphere and a strong nebular component of radiation (emission measure of ~1061 cm−3). The luminosity of the hot active object was estimated to > 5−8×103 Lʘ. Such high luminosity, sustained for the time of years, can be understood as a result of an enhanced transient accretion rate throughout a large disk, leading also to formation of collimated ejection from the WD