On Invariants Of Virtual Knots And Welded Knots

Virtual knot theory and welded knot theory are both a generalization of classical knot the-ory.Since Kauffman L.H.built up virtual knot theory in 1999,it has been developed by many topologists.In recent years,there has been tremendous interest in developing virtual knot theory.Currently,as a quotient theory of virtual knot theory,welded knot theory is becoming one of the hot issues in generalized knot theory.1.We study the properties of virtual Temperley-Lieb algebra and show how the f-poly-nomial of virtual knot can be derived from a representation of the virtual braid group into the virtual Temperley-Lieb algebra,which is an approach similar to Jones's original construction:.2.A certain class of welded knots K2n is considered.By calculating the commutators subgroup of fundamental group Gn of welded knot K2n,n ?Z+,we show that these welded knots are not equivalent to each other and they are all not classical knots.Secondly,we study some properties of Gn and obtain that Gn is linear,residually finite and hopfian.3.The unknotting number of a welded knot is considered.Firstly,we obtain an upper-bound of the unknotting number of a welded knot by using the warping degree method.Further,we introduce a standard welded torus knot with welded datum and obtain an upper bound of the unknotting number by an algorithm with warping degree method.we get a lower bound of the unkotting number of a welded knot by Alexander quandle colorings,we give a definition of Gordian distance for welded knots analogous to the classical case.Secondly,one result of Ma-Qiu about unknotting number of classical knot is extended to welded knot theory.Finally,we give a definition of a twist welded knot and consider its unknotting number and bridge number.4.We extend Adam's triple crossing number of a classical knot to the welded knot theory.We prove that every welded knot has a triple crossing projection and then investigate the triple crossing number C3(K)of a welded knot K,which is the minimum number of triple crossings in a projection of K,and obtain upper and lower bounds on C3(K)in terms of the traditional crossing number.Secondly,we extend the A-crossing number to welded knot theory.