On Gordian Complexes Of Some Local Moves Of Knots

Knot theory is a subject of the study of embeddings of circles in space.The fundamental problem of knot theory is the classification of knots up to ambient isotopy.The subject started to develop systematically in the late nineteenth century,and has been explosive growth during the last 40 years.In 1937,Wendt introduced the most well-known local move of knots,which we call the crossing change.Every link can be simplified to a trivial link by crossing changes.This is related to many results,for example,the construction of the Jones polynomial of links and the Reshetikhin-Turaev invariants of 3-manifolds.These invariants had a great impact on modern Knot Theory.Grossing change is a special kind of local moves.More generally,a local move of a link is a local modification of the link,in which a tangle T1 is replaced by another tangle T2.It turned out that results concerning local moves have important implication in the knot theory.In 2002,Hirasawa and Uchida introduced a simplicial complex called the Gordian complex of knots by the crossing change.In this paper,we study the Gordian complexes of other local moves of knots.The main contents are as follows:1.We constructed a new local move of knots,called#-move,and we study the properties of#-move.2.The properties of H(n)-Gordian complex are studied.We prove that for any knot K0,there exists an infinite family of knots {K0,K1,...} such that the#-move-Gordian distance d#(Ki,Kj)= 1 and H(n)-Gordian distance dH(n)(Ki,Kj)= 1 for any i ? j and all n ? 2.3.The properties of n-Gordian complex are investigated.We prove that if n is even,then for any knot K0',there exists an infinite family of knots {K0',K1;,…} such that the n-Gordian distance dn{Ki',Kj')= 1 and pass-move-Gordian distance dpass(Ki',Kj')= 1 for any i ? j.