On stick number of knots and links

The DNA strand is made up of small rigid sticks of sugar, phosphorus, nucleotide proteins and hydrogen bond. A cyclic molecule can be modelled by a sequence of sticks glued end-to-end so that the last one is also glued to the first. Thus we have mathematical stick knots or polygonal knots. This gives rise to some questions: For instance, what is the smallest number of atoms needed to construct a nontrivial knotted molecule? For such stick numbers, Negami gave an upper and lower bound. In the case when a given knot reaches its lower bound in Negami’s inequality, we gave a specific relation between both of its stick and crossing numbers. Hence we conclude that not one of the knots, at least, to 26 crossings can achieve its lower bound. In other words every knotted molecule of atoms up to 26 will have a number of bonds more than the lower bound in Negami’s inequality. Subsequently it is shown that some interesting class of torus knots does not achieve that lower bound in Negami’s inequality.