Abstract
Knot mosaics are used to model physical quantum states. The mosaic number of a knot is the smallest integer [Formula: see text] such that the knot can be represented as a knot [Formula: see text]-mosaic. In this paper, we establish an upper bound for the crossing number of a knot in terms of the mosaic number. Given an [Formula: see text]-mosaic and any knot [Formula: see text] that is represented on the mosaic, its crossing number [Formula: see text] is bounded above by [Formula: see text] if [Formula: see text] is odd, and by [Formula: see text] if [Formula: see text] is even. In the process, we develop a useful new tool called the mosaic complement.
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