Short: no
Long: a pseudo-terminal slave connection is one end of a connection, like a pipe. Both are ways that data is passed back and forth between user processes through the kernel.
Once one end is closed, you lose the connection. The ptsname description in POSIX says:
Upon failure, ptsname() shall return a null pointer. This could occur if fildes is an invalid file descriptor or if the slave device name does not exist in the file system.
Once you close a file descriptor, it is invalid.
If you want to reuse a connection, you could do some workaround such as passing the open slave file descriptor to a newly-created process, e.g., from a server application which you write to manage the slave file descriptors.
Further reading:
The above was written in 2016. A 2022 comment mentions TIOCGPTPEER, which is an ioctl code allowing an application to obtain a new file descriptor:
(since Linux 4.13) Given a file descriptor in fd that refers to a pseudoterminal master, open (with the given open(2)-style flags) and return a new file descriptor that refers to the peer pseudoterminal slave device. This operation can be performed regardless of whether the pathname of the slave device is accessible through the calling process's mount namespace.
(the closed file-descriptor is still invalid).