Solomon, a name which means 'peace', like the Hebrew shalem in 'Jerusalem', was the son of David and is accredited for building the first temple in Jerusalem about 1,000 B.C. (see 1 Kings 6). Most of what we know about King Solomon is found in the Bible, supplemented by a plethora of myths and magical legends.
(There are several Christian Crosses which incorporate variations of the Solomon Knot. See for example the Prince of Peace Cross.)
The Bible relates how Solomon, on becoming king, prayed to God for wisdom (1 Kings 3:7-15). Any lesser man would have prayed for health, wealth and happiness, but no, Solomon asked for a "discerning heart" and "to distinguish between right and wrong." And this is exactly what God gave Solomon (who was also later blessed with health, wealth and happiness).
One well-known story (1 Kings 3:16-28) is of two prostitutes who asked Solomon to resolve a dispute over which of them was the rightful mother of a baby. Solomon's answer was extreme, but couldn't have been fairer. He suggested using a sword to chop the baby in two, so that each woman could have half each. The true mother said she would rather the other woman take the child if that was the only way to save his life. Such compassion convinced Solomon that she was indeed the true mother and ruled that she should have the baby. Phew!
1 Kings 4:29-34 speaks more of Solomon's wisdom, and together with various other tales, Solomon is held up to have been the wisest of kings. The Solomon's Knot is therefore a symbol of wisdom.
There are several legends regarding the design of the Solomon's Knot symbol and each tale focuses on the interconnectivity of the two loops and their synergy. So in addition to wisdom, the symbol means strength.
Sometimes seen as a tattoo and because the symbol has two interlocking parts, it is also known as the Lover's Knot. (And before we get tied up further with the Solomon's Knot, here's a little story about a piece of string.)
Another story gave us the maxim: 'Together, we stand. Divided, we fall.' This saying is attributed to Æsop (620-560 BC) in his fable The Four Oxen and the Lion:
A Lion used to prowl about a field in which Four Oxen used to dwell. Many a time he tried to attack them; but whenever he came near they turned their tails to one another, so that whichever way he approached them he was met by the horns of one of them. At last, however, they fell a-quarrelling among themselves, and each went off to pasture alone in a separate corner of the field. Then the Lion attacked them one by one and soon made an end of all four.
Sad end for the oxen, but they proved their point.
Solomon's Knot has been used for centuries in art and mathematics. More recently (2006) chemists built a molecular Solomon's knot at the nanoscale (see http://www3.interscience.wiley.com/journal/113469124/abstract). And a molecule, however cleverly built, is nowhere near clever enough to out-wit a lion. (See also Albert and the Lion)
Interestingly, that scientific breakthrough came as a result of separating down problems into smaller and smaller issues that could be resolved. This follows the Divide and Conquer principle; the opposite of Divide and Fall. The achievement was academic rather than for any immediate practical application, but the formulae will remain on the shelf of Potentially Useful Knowledge waiting for the right problem to amble by.