The conjecture that all tie knots are alternating 2-bridge knots

A tie knot is a knot that can be represented by a tie diagram, as defined in the paper. A knot is alternating if it has an alternating diagram, and its bridge number is the minimum number of bridges among all diagrams of the knot, where a bridge is an arc containing at least one overcrossing.

Tie-knot conjecture. All tie knots are alternating 2-bridge knots.

The conjecture is motivated by the knot table through eight crossings: the knots absent from the tie knots are exactly the 3-bridge knots in that range, while the other listed tie knots are 2-bridge knots. The claim remains open according to the source.