It is proved that the coefficients of conway polynomial in z0,z1 and z2 and all ambient isotopic invariants. In particular,the coefficient in z2 of conway polynomial for link L with two components is only depend on sw...It is proved that the coefficients of conway polynomial in z0,z1 and z2 and all ambient isotopic invariants. In particular,the coefficient in z2 of conway polynomial for link L with two components is only depend on switch crossings in pairs.展开更多
A conformal structure of a prion protein is thought to cause a prion disease by S.B. Prusiner＇s theory. Knot theory in mathematics is useful in studying a topological difference of topological objects. In this articl...A conformal structure of a prion protein is thought to cause a prion disease by S.B. Prusiner＇s theory. Knot theory in mathematics is useful in studying a topological difference of topological objects. In this article, concerning this conjecture, a topological model of prion proteins （PrPc, PrPsc） called a prion-tangle is introduced to discuss a question of whether or not the prion proteins are easily entangled by an approach from the mathematical knot theory. It is noted that any prion-string with trivial loop which is a topological model of a prion protein can not be entangled topologically unless a certain restriction such as ＂Rotaxsane Property＂ is imposed on it. Nevertheless, it is shown that any two split prion-tangles can be changed by a one-crossing change into a non-split prion-tangle with the given prion-tangles contained while some attentions are paid to the loop systems. The proof is made by a mathematical argument on knot theory of spatial graphs. This means that the question above is answered affirmatively in this topological model of prion-tangles. Next, a question of what is the simplest topological situation of the non-split prion-tangles is considered. By a mathematical argument, it is determined for every n 〉 1 that the minimal crossing number of n-string non-split prion-tangles is 2n or 2n-2, respectively, according to whether or not the assumption that the loop system is a trivial link is counted.展开更多
An equivalent description for the torus knot is given in this paper, and the classification theorem of the torus knot is proved in an elementary method. Using the circular presentation of torus knot , we showed that t...An equivalent description for the torus knot is given in this paper, and the classification theorem of the torus knot is proved in an elementary method. Using the circular presentation of torus knot , we showed that the genus of the torus knot Kp,q is 1/2（p-1）（q-1） A knot called as bitorus knot is defined in the paper and showed . special that bitorus knot are all the connected sum of two torus knots.展开更多
文摘It is proved that the coefficients of conway polynomial in z0,z1 and z2 and all ambient isotopic invariants. In particular,the coefficient in z2 of conway polynomial for link L with two components is only depend on switch crossings in pairs.
文摘A conformal structure of a prion protein is thought to cause a prion disease by S.B. Prusiner＇s theory. Knot theory in mathematics is useful in studying a topological difference of topological objects. In this article, concerning this conjecture, a topological model of prion proteins （PrPc, PrPsc） called a prion-tangle is introduced to discuss a question of whether or not the prion proteins are easily entangled by an approach from the mathematical knot theory. It is noted that any prion-string with trivial loop which is a topological model of a prion protein can not be entangled topologically unless a certain restriction such as ＂Rotaxsane Property＂ is imposed on it. Nevertheless, it is shown that any two split prion-tangles can be changed by a one-crossing change into a non-split prion-tangle with the given prion-tangles contained while some attentions are paid to the loop systems. The proof is made by a mathematical argument on knot theory of spatial graphs. This means that the question above is answered affirmatively in this topological model of prion-tangles. Next, a question of what is the simplest topological situation of the non-split prion-tangles is considered. By a mathematical argument, it is determined for every n 〉 1 that the minimal crossing number of n-string non-split prion-tangles is 2n or 2n-2, respectively, according to whether or not the assumption that the loop system is a trivial link is counted.
文摘An equivalent description for the torus knot is given in this paper, and the classification theorem of the torus knot is proved in an elementary method. Using the circular presentation of torus knot , we showed that the genus of the torus knot Kp,q is 1/2（p-1）（q-1） A knot called as bitorus knot is defined in the paper and showed . special that bitorus knot are all the connected sum of two torus knots.
