Two Results For Twisted Links

Twisted knot theory,introduced by Bourgoin in 2008,is a generalization of virtual knot theory.In this paper,we generalize the results for virtual links to twisted links.Firstly,the Alexander Theorem and the Markov Theorem for virtual braids and links are important for understanding the structure and classification of virtual links.We first prove that any twisted link can be described as the closure of a twisted braid,which is unique up to certain basic moves.This is the analogue of the Alexander Theorem and the Markov Theorem for virtual links,respectively.Then we also give reduced presentations for the twisted braid group and the flat twisted braid group,which is mainly to explore braid generators and braid relationships with minimum number in the twisted braid group and the flat twisted braid group.Secondly,it is easily shown that any virtual knot can be deformed into a trivial knot by a finite sequence of generalized Reidemeister moves and two "forbidden moves"F1 and F2.Similarly,we show that any twisted knot also can be deformed into a trivial knot or a trivial knot with a bar by a finite sequence of extended Reidemeister moves and three"forbidden moves" T4,F1(or F2)and F3(or F4).