The Affine Index Polynomial And Writhe Polynomial Of Virtual Pure Tangle On Two Strands

The knot is an important object in geometrical topology theory.The classical knot is a smooth embedding of S~1in S~3(or R~3).The virtual knot is a generalization of the classical knot,which is defined as the smooth embedding of S~1in S_g×I.An important research direction in knot theory is to find a invariant under Reidemeister move transformation on the knot projection,which is called the knot invariant.Knot invariants provide important indexes for solving knot classification problems.Virtual pure tangle on two strands is an area of knots containing four input ends(includ-ing two input ends and two output ends),and there are no closed loops inside.In this thesis,according to the method of Nicolas Petit generalized Wriggle polynomial to virtual tangle,we extend affine index polynomial of virtual knot proposed by kauffman and writhe polynomial of flat long virtual knot proposed by Cheng-Gao to virtual pure tangle on two strands.Then we get the affine index polynomial and the writhe polynomial of the virtual pure tangle on two strands,and prove the equality of the two polynomials.Then a connected sum of long virtual knots is given,which can be proved not to be well defined by the polynomial in this thesis.