Fiberedness of almost-Montesinos knots

In this paper we begin to classify fiberedness of "Almost-Montesinos" knots, a generalization of Montesinos knots. We employ the method used in the classification of fiberedness of Montesinos knots due to Hirasawa and Murasugi. To achieve this classification, we find minimal-genus surfaces of "skew pretzel links" (a generalization of pretzel links) via sutured manifold decompositions, following Gabai's method for pretzel links. We end by stating three remaining cases.Mathematic

Extending fibrations on knot complements to ribbon disk complements

We show that if $K$ is a fibered ribbon knot in $S^3=\partial B^4$ bounding ribbon disk $D$, then with a transversality condition the fibration on $S^3\setminus\nu(K)$ extends to a fibration of $B^4\setminus\nu(D)$. This partially answers a question of Casson and Gordon. In particular, we show the fibration always extends when $D$ has exactly two local minima. More generally, we construct movies of singular fibrations on $4$-manifolds and describe a sufficient property of a movie to imply the underlying $4$-manifold is fibered over $S^1$.Comment: 59 pages, 51 figure

Knot cobordisms, bridge index, and torsion in Floer homology

Given a connected cobordism between two knots in the 3-sphere, our main result is an inequality involving torsion orders of the knot Floer homology of the knots, and the number of local maxima and the genus of the cobordism. This has several topological applications: The torsion order gives lower bounds on the bridge index and the band-unlinking number of a knot, the fusion number of a ribbon knot, and the number of minima appearing in a slice disk of a knot. It also gives a lower bound on the number of bands appearing in a ribbon concordance between two knots. Our bounds on the bridge index and fusion number are sharp for $T_{p,q}$ and $T_{p,q}\# \overline{T}_{p,q}$, respectively. We also show that the bridge index of $T_{p,q}$ is minimal within its concordance class. The torsion order bounds a refinement of the cobordism distance on knots, which is a metric. As a special case, we can bound the number of band moves required to get from one knot to the other. We show knot Floer homology also gives a lower bound on Sarkar's ribbon distance, and exhibit examples of ribbon knots with arbitrarily large ribbon distance from the unknot.Comment: 21 pages, 7 figures, to appear in the Journal of Topolog

Explicitly describing fibered 3-manifolds through families of singularly fibered surfaces

We give an explicit description of a fibration of the complement of the closure of a homogeneous braid, understanding how each fiber intersects every cross-section of $S^3$.Comment: 19 pages, 18 figures. Written for the proceedings of "Frontiers in Geometry and Topology," a summer 2022 conference in honor of Tomasz Mrowka's 60th birthda

A Preference for Freedom: Kantian Implications for an Incompatibilist Will and Practical Accountability

This thesis aims to provide a coherent account of free will and practical grounds to prefer it. Its goal is to develop a pragmatic understanding of agency by which to hold individuals morally accountable. The paper begins with a critique of P.F. Strawson, whose seminal paper “Freedom and Resentment” bypasses the question of free will altogether in its claims about morality. Subsequently, it proceeds to a defense of incompatibilism that traces an argument through the existing literature. From this position, it claims that neither Strawson nor traditional compatibilists can provide an account of morality that is reliable or well enough defined to play the role required of it. Instead of being left with hard determinism, however, Kant opens the door to a metaphysics that exists outside of our epistemological limits. Rather then derive an account based on this metaphysics, the necessary characteristics of a free will are derived from an account of morality and proven to be possible using Kantian epistemology. The paper concludes by positing three distinct reasons to prefer a free will framework to a deterministic framework, provided our inability to answer the question empirically. These draw on Pascal’s Wager, William James’ “The Will to Believe,” and inference to the best explanation