| A quadrisecant line for a knot K is defined as a line ℓ in R3 which intersects K in at least four distinct points a,b,c,d, in that order along ℓ, no two of which lie in a common straight sub-arc of K. We say that the 4-tuple of points abcd is a quadrisecant for K. There are several types of quadrisecants in a knot: simple, flipped and alternating quadrisecants.;The main results of this dissertation is to give a lower bound for the total number of trisecants in a knot in terms of its crossing number. E. Denne proves that any non-trivial knot contains an essential alternating quadrisecant ([Den05]). Using our result, we conjecture that a knot K with crossing number, cr(K), has at least 1/2 2crK +132 quadrisecants.;The ropelength of a knot K is defined as the shortest length of unit radius rope needed to tie the knot K. The existence of an essential alternating quadrisecant has been used to give a lower bound for the ropelength of knots ([DDS06]). In this dissertation we explore an extension of this method using the existence of an essential alternating quadrisecant to find a better upper bound for the ropelength of a trefoil knot. So far we have found a piece-wise smooth trefoil knot with length 32.4777 around which we can place a tube of radius one around each individual smooth piece of the knot. In order to give an upper bound, we must give a careful estimate of the largest thickness of a tube associated with a smoothing of a knot at the points at which the knot fails to be a C1 curve. We expect that the ropelength of this smoothed trefoil will increase slightly or, equivalently, the injective radius will decrease slightly near these points. In order to get a new upper bound, the total length cannot be more than 32.7431, the smallest upper bound for a polygonal approximation via simulations in [BPR05]. |
