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Triple Crossing Number of Knots and Links

A triple crossing is a crossing in a projection of a knot or link that has three strands of the knot passing straight through it. A triple crossing projection is a projection such that all of the crossings are triple crossings. We prove that every knot and link has a triple crossing projection and then investigate c_3(K), which is the minimum number of triple crossings in a projection of K. We obtain upper and lower bounds on c_3(K) in terms of the traditional crossing number and show that both are realized. We also relate triple crossing number to the span of the bracket polynomial and use this to determine c_3(K) for a variety of knots and links. We then use c_3(K) to obtain bounds on the volume of a hyperbolic knot or link. We also consider extensions to c_n(K).