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l. kauffman @xcite introduced the notion of virtual knots and links , which is a generalization of classical knots and links .
virtual links are in one - to - one correspondence to abstract links on oriented surfaces @xcite , and to stable equivalence classes of links in oriented thickened surfaces over oriented surfaces @xcite .
m. bourgoin @xcite generalized them to twisted links .
twisted links are in one - to - one correspondence to abstract links on ( possibly non - orientable ) surfaces , and to stable equivalence classes of links in oriented thickened surfaces over ( possibly non - orientable ) surfaces @xcite .    in this paper , we introduce the notion of the double covering of a twisted link ( section  [ sect : doublea ] ) .
it is defined by considering the orientation double covering of an abstract link .
we give an explicit method of constructing a virtual link diagram @xmath0 from a twisted link diagram @xmath1 , called a double covering diagram , such that the virtual link @xmath2 $ ] is the double covering of the twisted link @xmath3 $ ] ( section  [ sect : doubleb ] ) .    as an application
, one can obtain an invariant of twisted links from an invariant of virtual links by considering the double coverings .
for example , bourgoin s twisted link group @xcite is understood as the virtual link group of the double covering .
this gives a geometric interpretation of the twisted link group , which answers the problem proposed by bourgoin in @xcite .
similarly the twisted link quandle of a twisted link defined by the first author in @xcite is the ( upper ) virtual link quandle of the double covering .
these are discussed in section  [ sect : groups ] .
we also discuss links in oriented thickened surfaces , their diagrams and stable equivalence classes in sections  [ sect : thickeneda ] and  [ sect : thickenedb ] .
there is a bijection between the set of stable equivalence classes of links in oriented thickened surfaces over closed surfaces ( or equivalently the set of stable equivalence classes of generalized link diagrams over closed surfaces ) and the set of abstract links .
a _ virtual link diagram _ is a link diagram in @xmath4 which may have virtual crossings ; a _
virtual crossing _ is an encircled double point without over - under information .
a _ virtual link _ is the equivalence class of a virtual link diagram where the equivalence relation is generated by isotopies of @xmath4 , reidemeister moves ( fig .
[ fig : movesr ] ) and virtual reidemeister moves ( fig .
[ fig : movesv ] ) .
a _ twisted link diagram _ is a link diagram which may have virtual crossings or bars ; a _ bar _ is a small edge intersecting transversely with an arc of the diagram .
twisted link _ is the equivalence class of a twisted link diagram where the equivalence relation is generated by isotopies of @xmath4 , reidemeister moves , virtual reidemeister moves and twisted reidemeister moves ( fig .
[ fig : movest ] ) .
reidemeister moves , virtual reidemeister moves and twisted reidemeister moves are called r - moves , v - moves and t - moves for short . a virtual link
( or a twisted link ) with one component is called a _ virtual knot _ ( or a _ twisted knot _ ) .       for a positive crossing @xmath5 with labels as in fig .  [
fig : labels ] ( i ) , we consider 4 relations : @xmath6 the first two relations @xmath7 is denoted by @xmath8 and the other two relations @xmath9 is denoted by @xmath10 .    for a negative crossing @xmath5 with labels as in fig .
[ fig : labels ] ( ii ) , we consider 4 relations : @xmath11 the first two relations @xmath12 is denoted by @xmath8 and the other two relations @xmath13 is denoted by @xmath10 .
the _ twisted link group _ of @xmath1 , denoted by @xmath17 , is the group determined by a group presentation whose generators are the symbols @xmath18 and @xmath19 @xmath20 and the defining relations are the 4 relations for each crossing and the 2 relations for each bar .        proof . without loss of generality
we may assume that @xmath1 is in @xmath21 and in general position with respect to the height function @xmath22 .
let @xmath23 be the semi - arcs of @xmath1 .
by definition , the twisted link group @xmath17 of @xmath1 has a presentation whose generators are @xmath24 and the defining relations are @xmath25    let @xmath26 and @xmath27 be the diagrams obtained from @xmath1 and @xmath28 by cutting arcs at all bars .
( although the diagram @xmath26 has some end - points , the ( upper ) virtual link group can be defined even for such a diagram . )
the virtual link group @xmath29 of @xmath26 has a presentation @xmath30 and the virtual link group @xmath31 of @xmath27 has a presentation @xmath32 the diagram @xmath0 is obtained from the union of @xmath26 and @xmath27 by connecting their endpoints with some arcs , and they make additional relations @xmath16 for all bars @xmath14 . thus the virtual link group @xmath33 is the amalgamated free product of @xmath29 and @xmath31 with amalgamation by using @xmath16 for all bars @xmath14 .
this coincides with the twisted link group @xmath17 .
[ thm : twistedgroupb ] let @xmath1 be a twisted link diagram and let @xmath0 be a double covering diagram of @xmath1 .
let @xmath34 be a link in the oriented thickened surface @xmath35 presented by the ordinary abstract link diagram @xmath36 .
then the twisted link group @xmath37 is isomorphic to the fundamental group of the complement of @xmath34 in the singular @xmath38-manifold @xmath39 .    proof . by theorem  [ thm : twistedgroupa ]
, the twisted link group @xmath37 is isomorphic to the virtual link group @xmath33 of @xmath0 .
since @xmath0 is a virtual link diagram , it is shown in @xcite that the virtual link group @xmath33 is isomorphic to the fundamental group of the complement of @xmath34 in @xmath40 .
we may assume that @xmath1 is in @xmath21 and in general position with respect to the height function @xmath22 .
let @xmath23 be the semi - arcs of @xmath1 .
let @xmath26 and @xmath27 be the diagrams obtained from @xmath1 and @xmath28 by cutting arcs at all bars .
the lower virtual link group @xmath42 of @xmath26 has a presentation @xmath43 and the lower virtual link group @xmath44 of @xmath27 has a presentation @xmath45 the lower virtual link group @xmath41 is the amalgamated free product of @xmath42 and @xmath44 with amalgamation by using @xmath16 for all bars @xmath14 .
thus @xmath41 has the same group presentation with @xmath33 .
the first author introduced the twisted link quandle of a twisted link ( @xcite ) . for a twisted link diagram @xmath1 , the _ twisted link quandle _
@xmath47 is the quandle determined by a quandle presentation obtained from the group presentation of the definition of @xmath48 by changing each conjugate @xmath49 by the quandle operation @xmath50 and @xmath51 by the dual operation @xmath52 .
then the twisted link quandle is an invariant of a twisted link ( @xcite ) .
[ thm : twistedquandleb ] let @xmath1 be a twisted link diagram and let @xmath0 be a double covering diagram of @xmath1 .
let @xmath34 be a link in the oriented thickened surface @xmath35 presented by the ordinary abstract link diagram @xmath36 .
then the twisted link quandle @xmath47 is isomorphic to the fundamental quandle of @xmath34 in the singular @xmath38-manifold @xmath39 .    proof . by theorem  [ thm : twistedquandlea ]
, the twisted link quandle @xmath47 is isomorphic to the ( upper ) virtual link quandle @xmath53 of @xmath0 .
since @xmath0 is a virtual link diagram , it is shown in @xcite that the virtual link quandle @xmath53 is isomorphic to the fundamental quandle of @xmath34 in @xmath40 .
bourgoin @xcite proved that if @xmath1 is a virtual link diagram , then @xmath55 is the free product of the ( upper ) virtual link group @xmath56 and the lower virtual link group @xmath57 .
this fact is also seen from the proof of theorem  [ thm : twistedgroupa ] .
similarly , from the proof of theorem  [ thm : twistedquandlea ] , we obtain the following .
twisted links are a generalization of virtual links . 
 as virtual links correspond to abstract links on orientable surfaces , twisted links correspond to abstract links on ( possibly non - orientable ) surfaces . in this paper , we introduce the notion of the double covering of a twisted link . 
 it is defined by considering the orientation double covering of an abstract link or alternatively by constructing a diagram called a double covering diagram . 
 we also discuss links in thickened surfaces , their diagrams and their stable equivalence classes . 
 bourgoin s twisted knot group is understood as the virtual knot group of the double covering .    
 keywords : twisted link , virtual link , abstract link , double covering , orientation double covering , thickened surface , generalized link diagram , twisted link group +