 Together, with Olafur Thordarson, Thordur Hognason, Magnus Einarsson and later Bjarni Sveinbjornsson, they formed the Kuran Swing a band that played jazz music influenced by Django Reinhardt.The band, that was only going to play for a single concert in 1989, was active for 13 years, performed in America and Europe, and "Kuran Swing" music to my ears (2000)

In 1993, Björn formed a band that arranged Icelandic children's songs into a modern jazz style and released the album "Göngum svo léttir" ("Walking so lightly".In 1997, Björn made an album mostly with jazz standards, BT Jazz Guitar, with fellow guitarists; Philip Catherine, Dough Raney, Jacob Fisher, Paul Weeden and Leivur Thomsen.