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This note explores the use of Newton polytopes in the study of Lagrangian fillings of Legendrian submanifolds. In particular, we show that Newton polytopes associated to augmented values of Reeb chords can distinguish infinitely many distinct Lagrangian fillings, both for Legendrian links and higher dimensional Legendrian spheres. The computations we perform work in finite characteristic, which significantly simplifies arguments and also allows us to show that there exist Legendrian links with infinitely many nonorientable exact Lagrangian fillings. [ABSTRACT FROM AUTHOR]

A multi-crossing (or n-crossing) is a singular point in a projection of a knot or link at which n strands cross so that each strand bisects the crossing. We generalise the classic result of Kauffman, Murasugi and Thistlethwaite relating the span of the bracket polynomial to the double-crossing number of a link, span〈K〉 ⩽ 4c2, to the n-crossing number. We find the following lower bound on the n-crossing number in terms of the span of the bracket polynomial for any n ⩾ 3: $$\text{span} \langle K \rangle \leq \left(\left\lfloor\frac{n^2}{2}\right\rfloor + 4n - 8\right) c_n(K).$$ We also explore n-crossing additivity under composition, and find that for n ⩾ 4 there are examples of knots K1 and K2 such that cn(K1#K2) = cn(K1) + cn(K2) − 1. Further, we present the the first extensive list of calculations of n-crossing numbers of knots. Finally, we explore the monotonicity of the sequence of n-crossings of a knot, which we call the crossing spectrum. [ABSTRACT FROM AUTHOR]

An n-crossing is a point in the projection of a knot where n strands cross so that each strand bisects the crossing. An übercrossing projection has a single n-crossing and a petal projection has a single n-crossing such that there are no loops nested within others. The übercrossing number, ü(K), is the smallest n for which we can represent a knot K with a single n-crossing. The petal number is the number of loops in the minimal petal projection. In this paper, we relate the übercrossing number and petal number to well known invariants such as crossing number, bridge number, and unknotting number. We find that the bounds we have constructed are sharp for (r, r + 1)-torus knots. We also explore the behavior of übercrossing number under composition. [ABSTRACT FROM AUTHOR]