A bow tie (or dickie bow (PROSE: A Big Hand for the Doctor) was a form of necktie in the shape of a bow. They could be either pre-tied or self-tied. The Doctor wore a bow tie in many of his incarnations, though the traditional bow tie was most often associated with the Second, Third and Eleventh Doctors.
The Doctor's bow ties
The Second Doctor's bow tie appeared along with his other clothes when his first incarnation regenerated with the TARDIS' assistance. As with the rest of his clothing, the Second Doctor's bow tie was untidy and crumpled; it was a small pre-tied dark tie with small spots, attached to his collar with a safety pin. (TV: The Power of the Daleks et al.)
The Third Doctor wore a variety of bow ties towards the latter part of his life. These were chosen to match his various outfits, and he had several in different sizes and colours. (TV: The Three Doctors, Frontier in Space, et al.)
The Eleventh Doctor adopted a bow tie along with the other clothes he stole from Royal Leadworth Hospital soon after his regeneration. (TV: The Eleventh Hour) He was extremely fond of his bow tie and would often insist that "bow ties are cool" in spite of his companions' criticism. (TV: The Eleventh Hour, Vincent and the Doctor, The Lodger, A Christmas Carol, et al.) He also included a white bow tie as part of his formal wear, seen during Amy and Rory's wedding (TV: The Pandorica Opens) and on the occasion of his death. (TV: Let's Kill Hitler) During his period of retirement after losing Amy Pond and Rory Williams, the Eleventh Doctor changed his outfit to something more befitting his sojourn in Victorian London, but put on a bow tie again when spurred back into adventure by Clara Oswin Oswald. (TV: The Snowmen) He wore both classic and clip-on bow ties and told Alice Obiefune that he wore clip-ons because someone had once tried to use his bow tie to throttle him. (TV: The Bells of Saint John, COMIC: The Friendly Place) Before his regeneration into the Twelfth Doctor, he removed it and let it drop to the floor as a sign of his upcoming change. (TV: The Time of the Doctor)
After being visited in childhood by the Eleventh Doctor and assured that "bow ties are cool", Kazran Sardick wore a bow tie throughout his life. Kazran's father, Elliot Sardick, was also pictured wearing a bow tie. (TV: A Christmas Carol)
Having received a glimpse of his eleventh incarnation through his psychic paper in 2724, the Second Doctor described his bow tie as "rather fetching". (AUDIO: Shadow of Death) By contrast, when the Fourth Doctor witnessed a statue of his future self, he expressed disbelief at his future incarnation's choice of attire. (PROSE: The Roots of Evil) The Twelfth Doctor also dismissed bow ties as "embarrassing". (TV: Time Heist) However, upon seeing Adrian Davies, a teacher at Coal Hill School with a fashion sense reminiscent of his previous incarnation, complete with bow tie, he responded more favourably because he was under the mistaken impression that he was the boyfriend of his companion Clara Oswald, taking pride in that she was dating a man resembling him. (TV: The Caretaker)
He would later compliment Osgood on her "nice bow tie", which she took to wearing after meeting his predecessor, to which she quoted him "bow ties are cool". Osgood's bow tie would later be disintergrated along with herself by Missy. (TV: Death in Heaven) However, it later became unclear if the Osgood that got disintegrated was the actual Osgood or her Zygon double, (TV: The Day of the Doctor, The Zygon Invasion / The Zygon Inversion) which meant the bow tie could have been part of a camouflage and not actual clothing.
The Master wore a midnight blue bow tie when he met the Seventh Doctor, Ace, and the Cheetah People. (TV: Survival) He also wore a black bow tie when posing as O at Daniel Barton's party. (TV: Spyfall)