Biquandles, semi-Wirtinger number and bridge index
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The bridge index is a useful invariant of knots. For classical knots, there is no ambiguity in its definition. For virtual knots, however, multiple natural definitions of the bridge index exist. To demonstrate that the difference between the two formulations can be arbitrarily large, we introduce a quantity called the semi-Wirtinger number of a virtual knot. A function of our numerical complexity turns out to also provide an upper bound for the number of biquandle homomorphisms from fundamental biquandles to other biquandles and offers a lower bound for the second formulation of the bridge index and hyperbolic volume of some classes of knots.
Abstract
The bridge index is a useful invariant of knots. For classical knots, there is no ambiguity in its definition. For virtual knots, however, multiple natural definitions of the bridge index exist. To demonstrate that the difference between the two formulations can be arbitrarily large, we introduce a quantity called the semi-Wirtinger number of a virtual knot. A function of our numerical complexity turns out to also provide an upper bound for the number of biquandle homomorphisms from fundamental biquandles to other biquandles and offers a lower bound for the second formulation of the bridge index and hyperbolic volume of some classes of knots.