Lower and upper bounds on edge numbers and crossing numbers of knots

Special types of functions called knot invariants are used to distinguish between the infinitely many distinct knot types. Edge number {dollar}e(K){dollar} and crossing number {dollar}c(K){dollar} are two elementary knot invariants which are examined in detail in this thesis. The knot invariant {dollar}e(K){dollar} specifies the minimal number of edges required to form a knot of type K, and {dollar}c(K){dollar} is the minimal number of crossings which appear in a regular projection of K. Another knot invariant which plays an important role in this thesis is {dollar}csb2(K),{dollar} the coefficient of {dollar}zsp2{dollar} in the Conway polynomial of K.; Although edge number and crossing number are easily defined, relatively few results involving these invariants are known. The focus of this thesis is on the determination of improved bounds on the values of these invariants for both prime and composite knots. Upper bounds on edge numbers of several prime knots are obtained and piecewise-linear illustrations of these knots along with integer coordinates for the vertices are also presented.; It is not known in general whether {dollar}e(Ksb1{dollar}#{dollar}Ksb2) > e(Ksb1){dollar}. Conditions implying that this inequality holds are determined as well as lower bounds on the value of n so that the composite knot (#K) {dollar}sbsp{lcub}i=1{rcub}{lcub}n{rcub}{dollar} has edge number greater than e(K). Similar results involving crossing numbers are also obtained.; The main result of this thesis states that {dollar}vert csb2(K)vert le 6{dollar} whenever {dollar}e(K)le 8.{dollar} Thus, if {dollar}vert csb2(K)vertge 7,{dollar} then {dollar}e(K)ge 9{dollar}. Since it is known that {dollar}e(K)ge 9{dollar} whenever {dollar}c(K)ge 14{dollar}, an improved lower bound of nine has been determined for the edge number of all knots for which {dollar}vert csb2(K)vert ge7{dollar} and {dollar}e(K)<14{dollar}.