I once wrote something extensive on Pontius and Hippolytus (and Cassian) which I will reproduce here for your convenience (edited):
Today is the commemoration of Sts. Hippolytus and Cassianus (or Cassian), martyrs (13 August). “But, Father! Wait!”, you are sure to being saying. In the 1970MR on 13 August we find the feast of Sts. Hippolytus, priest and Pontianus, Pope, martyrs, not Cassian at all! In the 2004 Martyrologium Romanum (MartRom – p. 449) we find not only the listing for Hippolytus with Pontianus (or Pontian) but also, by himself, Cassian. Since Cassian and Hippolytus had nothing whatsoever to do with each other, after the reform of the liturgy Hippolytus was logically put together with Pontianus on the same feast. There are actually quite a few ancient saints by name of Hippolytus and there is a lot of confusion about which is which. This one is certainly a Hippolytus of Rome and not the prolific writer of the same name who, according in P. Nautin’s article in the Encyclopedia of the Early Church (vol. I, pp. 383-4), was a bishop in Palestine who died after A.D. 240.
The third century was a turbulent time for Christians in Rome. Not only were there persecutions from without, there were also theological controversies within the Church about the nature(s) and divinity of Christ and His relationship with the Father. Hippolytus, who wrote in Greek, was a pivotal figure in the early Roman Church. Among other things, he championed a position against Modalism, which idea was that the Persons of the Trinity were merely three modes or manifestations of one God without being individual Persons. Hippolytus forwarded the idea that Christ the Logos was so separate from the Father, though subordinate to Him, that Christ virtually was another God (Ditheism). Pope Zephyrinus would not make a firm statement one way or another and Hippolytus condemned him as the weak puppet of the powerful deacon Callistus. Zephyrinus in 217 or 218 exits the stage and Callistus was elected to the See of Peter. Hippolytus then got himself elected “pope” by his followers. A terrible rigorist, he forthwith accused Callistus of various heresies and laxity in ecclesiastical discipline. He was thus antipope during the reigns of Callistus I, Urban I (222-230) and our man Pontianus (230–235) who according to legend is sometimes credited (more than likely erroneously) with introducing the liturgical greeting and response and hitherto translation bugbear of liturgists “Dominus vobiscum… Et cum spiritu tuo.”
During the persecution by Maximinus Thrax in 235, Pope Pontianus and the priest/antipope Hippolytus were condemned ad metalla (“to the mines”) and banished to Sardinia, called an unhealthy island (insula nociva). Pontianus probably renounced his office on 28 September, according to ancient sources. So, Pope Celestine V (5 July 1294 (crowned 29 August) – 13 December 1294 and died 19 May 1296) was not be the only Pope in history to have resigned. Perhaps together in the terrible mines of Sardinia or en route, Hippolytus and Pontianus were reconciled before they died. Pope Fabian (236-250) had their bodies brought back to Rome. They are feasted on the same day probably because in the IV c. document concerning the interment of martyrs, the Depositio Martyrum, we read “Idus Aug. Ypoliti in Tiburtina et Pontiani in Callisti”. In August the Ides are on the 13th day. Pontianus was buried in the papal section of the catacomb of Callistus and Hippolytus on the Via Tiburtina. The fact that they were both so venerated by the Romans is held as a proof that they were reconciled with each other.
Cassian, on the other hand, according to the hymn of Prudentius (cf. Peristephanon, Hymn IX), was a teacher at Forum Cornelii (named after the dictator L. Cornelius Sulla – modern Imola). He was handed over (c. 300 according to 2001 MartRom) to his pupils who tortured him to death using their writing styluses (traditus est calamis ad mortem torquendus), made of iron, reeds or other pointy hard materials with which they would draw on wax tablets, etc. The MartRom adds a note that Cassian was given to his students to be killed because, “the weaker the hand, the more painful was the sentence of martyrdom.” Today, students torture their teachers to death with PDA styluses, laptop computers and MP3 players, not to mention execrable English – but I digress.