The arm ring and its significance
Forged by the same dwarven smiths that made Mjölnir and the great boar Gullinbursti, Odin himself was gifted a ring of solid gold named "Draupnir".
This ring would multiply itself eight times over every ninth night, and it may actually be the inspiration for the importance and sacred nature of the Viking Arm Ring that would become an integral part of the Norse culture.
Vikings had a culture fixated on honor. Many of their laws focused on upholding honor or punishment for those who broke their oath. Even to get to Valhalla, one needed to die honorably in battle.
Deity
Obviously one of the most binding ways to swear an oath would be to swear upon the Aesir themselves.
From the Landnámabók, a medieval book of Icelandic law, we see examples of swearing on rings to the Aesir and Vanir themselves. Specifically mentioned are Njörðr, Freyr, and a third figure, whose title translates to “Almighty God”, which could refer to several different deity, but most likely Odin himself.
Weapons
Other historical documents from the time cite Vikings swearing oaths upon weapons, which is a logical thing to do when you consider the culture.
In essence it’s saying “I will uphold my oath, because if I do not, I will not enter Vahlalla”. It would be an oath difficult to break, and one which would certainly bind both parties to honor.
The Arm Ring
This brings us to the arm ring itself. The Norse put great value on jewelry and adornment, and even used jewelry as a status symbol, decorating themselves, their wives, and their families to show how much wealth they had accumulated.
Nonetheless, the arm ring held a different significance; if deity and weapons were important things to swear upon, the arm ring must have been something which tied all possible oaths together.
Modern Application
While there are far fewer instances in which one might swear an oath on an arm ring, it is still a relevant symbol. An arm ring today may represent a commitment to one's own values, to a spouse, to honor and integrity. And while you may not be making an oath to a king, maybe it will inspire that same commitment in your day to day oaths!