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  1. ; (, Geometriae Dedicata)
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  2. ; ; ; ; ; ; ; ; ; ; et al (, Journal of Knot Theory and Its Ramifications)
    Meier and Zupan proved that an orientable surface [Formula: see text] in [Formula: see text] admits a tri-plane diagram with zero crossings if and only if [Formula: see text] is unknotted, so that the crossing number of [Formula: see text] is zero. We determine the minimal crossing numbers of nonorientable unknotted surfaces in [Formula: see text], proving that [Formula: see text], where [Formula: see text] denotes the connected sum of [Formula: see text] unknotted projective planes with normal Euler number [Formula: see text] and [Formula: see text] unknotted projective planes with normal Euler number [Formula: see text]. In addition, we convert Yoshikawa’s table of knotted surface ch-diagrams to tri-plane diagrams, finding the minimal bridge number for each surface in the table and providing upper bounds for the crossing numbers. 
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  3. ; ; ; (, Pacific Journal of Mathematics)
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  4. ; (, Topology proceedings)
    We examine the Kauffman bracket expansion of the generalized crossing $$\Delta_n$$, a half-twist on $$n$$ parallel strands, as an element of the Temperley-Lieb algebra with coefficients in $$\mathbb{Z}[A,A^{-1}]$$. In particular, we determine the minimum and maximum degrees of all possible coefficients appearing in this expansion. Our main theorem shows that the maximum such degree is quadratic in $$n$$, while the minimum such degree is linear. We also include an appendix with explicit expansions for $$n$$ at most six. 
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