Unknotting Numbers Of Several Infinite Families Of Link Diagrams

The essential problem in Knot Theory is how to distinguish distinct knots or links,while the knot invariants are major tools for judging whether or not two knots or links are equivalent.There are many knot invariants:such as the unknotting number,the braid index,the crossing number,the bridge number,knot polynomials and the genus.Among them,unknotting number of a non-trivial knot K is the minimal number of crossing changes required to transform K into the unknot.The methods used to determine the unknotting number of a knot are mainly the Jones Polynomial,the Heegaard Floer homology,the R-moves,the isotopy and the Alexander Polynomial Surgery.As so far the research results on the unknotting number are relatively limited.In this paper,some new certain types of Montesinos link diagrams are studied,and the unknotting numbers have been determined.The main results are as follows:The paper first gives 23 kinds trivial specific expressions of link diagrams in {M(TX1,TX2,Tx3,X4)}.According to the parity of crossing numbers,we classify the non-trivial link diagrams in {M(TX1,TX2,TX3,±2),sgn(X1)=sgn(X2)},and then we divide them into certain families.For every non-trivial link diagram D,obtain the number u of crossing changes required to transform it into trivial.Furthermore,when any number of crossing changes is less than u,one can not transform D into trivial.Then the unknotting num-ber of D is u.Inspired by this method,extend to link diagrams in {M(TX1,TX2,TX3,X4)}.According to the trivial specific expressions of link diagram in {M{TX1,TX2,TX3,X4)} and Reduction Crossing,we calculate the unknotting number of link diagram in it.