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  1. ; ; ; ; (, Mathematische Annalen)
    Abstract We prove an equivariant version of the Cosmetic Surgery Conjecture for strongly invertible knots. Our proof combines a recent result of Hanselman with the Khovanov multicurve invariants$${\widetilde{{{\,\textrm{Kh}\,}}}}$$ Kh ~ and$${\widetilde{{{\,\textrm{BN}\,}}}}$$ BN ~ . We apply the same techniques to reprove a result of Wang about the Cosmetic Crossing Conjecture and split links. Along the way, we show that$${\widetilde{{{\,\textrm{Kh}\,}}}}$$ Kh ~ and$${\widetilde{{{\,\textrm{BN}\,}}}}$$ BN ~ detect if a Conway tangle is split. 
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  2. ; ; ; ; (, Transactions of the American Mathematical Society, Series B)
    We show that if a composite θ-curve has (proper rational) unknotting number one, then it is the order 2 sum of a (proper rational) unknotting number one knot and a trivial θ-curve. We also prove similar results for 2-strand tangles and knotoids. 
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    Free, publicly-accessible full text available March 4, 2026
  3. ; ; (, The Art of Discrete and Applied Mathematics)
    Bootstrap percolation is a process defined on a graph which begins with an initial set of infected vertices. In each subsequent round, an uninfected vertex becomes infected if it is adjacent to at least r previously infected vertices. If an initially infected set of vertices, A0, begins a process in which every vertex of the graph eventually becomes infected, then we say that A0 percolates. In this paper we investigate bootstrap percolation as it relates to graph distance and connectivity. We find a sufficient condition for the existence of cardinality 2 percolating sets in diameter 2 graphs when r = 2. We also investigate connections between connectivity and bootstrap percolation and lower and upper bounds on the number of rounds to percolation in terms of invariants related to graph distance. 
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