On the uniform thickness property and contact geometric knot theory
LaFountain, Douglas J.
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This work is an investigation of the uniform thickness property (UTP) for knots in the contact 3-sphere. We establish cabling theorems for the UTP: in particular we show that if a knot type K satisfies the UTP, then cablings K ( P,q ) also satisfy the UTP; we also show that if K is a χ-candidate knot type (for potential failure of the UTP), then positive cablings K ( P,q ) are also χ-candidate knot types. This leads us to provide a complete UTP classification for the class of iterated torus knots, namely showing that an iterated torus knot K r = (( P 1 , q 1 ),···, ( P r , q r )) fails the UTP if and only if P i > 0 for all i. More specifically, we identify all non-thickenable solid tori in the class of iterated torus knots which result in failure of the UTP. We then are able to show that failure of the UTP in the class of iterated torus knots is a sufficient condition for the existence of transversally non-simple cablings. We also identify large families of Legendrian simple iterated torus knots, including many that are cablings of knots that fail the UTP. We then establish cabling theorems for positively twisting, interlocking, steps configurations, and use this to show that every non-thickenable solid torus in the class of iterated torus knots can be represented by such an interlocking steps configuration. This then allows us to establish new knot-type specific Legendrian and transversal Markov Theorems without stabilization for the ((2, 3), (1, 2)) iterated torus knot.