the rulers were nominally or actually vassals of the King of Denmark, including Jarl Haakon. It is undisputed that later kings, until Magnus IV (Magnus the Blind, r. 1130–1135 and 1137–1139), were descended from Harald Hardrada: the 'Hardrada dynasty'. However, many modern historians doubt whether Harald III or his predecessors Olaf Tryggvason, Olaf II and Magnus the Good were in fact descended from Harald Fairhair (for example questioning the identification of Halfdan in Hadafylke with the father of Sigurd Syr or Harald Fairhair's fathering of Sigurd Hrise on a Sami girl called Snæfrithr Svásadottir) and whether they in fact made such a claim, or whether these lineages are a construction from the 12th century. Sverre Sigurdsson's claim to be the son of Sigurd Munn is also usually considered to be dubious, which would make Inge II (Inge Bårdsson) possibly the last king of a dynasty. Scholars now consider the Fairhair dynasty at least partly the product of medieval invention. One motive would be to increase the legitimacy of rulers by giving them a clear royal ancestry dating back to the foundation of the kingdom. Another was to provide pedigrees for other people by connecting them to the royal house. Versions of the royal descent are preserved in various works by Icelandic skalds and historians, some based on now lost works: Þjóðólfr of Hvinir's Ynglingatal, in Nóregs konungatal (which preserves information from a lost work by Sæmundr fróði), and at greatest length in Snorri Sturluson's Heimskringla (which preserves information from a now lost version of Ari Þorgilsson's Íslendingabók). These differ in some respects. Joan Turville-Petre explored the relationship between them and argued that the original aims were to establish a framework of regnal years for dating and to connect Icelandic chieftains to them, and that the Vestfold origin of the dynasty was deliberately altered and they were connected to the Swedish Ynglings rather than the Skjǫldungs to fit Icelandic tradition. Claus Krag argued that an important motive was to establish a hereditary claim to Viken, the region around Oslo, because the area had been paying taxes to the King of Denmark. Turville-Petre speaks of a "decisive reconstruction of Harald [Hardrada]'s ancestry probably carried out by Icelanders, some two hundred years after his time" which made Halfdan the Black the progenitor of a dynasty which stretched in three branches from Harald Fairhair to Olaf Tryggvason, Olaf II and Harald. - in fulfillment