Lock Tie Knot

Lock Tie Knot is a puzzle game about the difference between a tangled rope and a genuine knot. A closed rope is shown on a dark canvas, drawn with warm amber light. Where one part of the rope passes over another, the under-strand shows a small gap. These are crossings — the only information that distinguishes one knot projection from another. Small indicators pulse at certain crossings. The player taps one and selects a move. The rope rearranges. Some crossings disappear. Others remain. The moves are Reidemeister moves — three specific local transformations proven by Kurt Reidemeister in 1927 to be sufficient for simplifying any knot diagram. Move I removes a single self-crossing by untwisting a loop. Move II removes two crossings simultaneously by sliding two parallel strands apart. Move III repositions crossings within a triangular region without removing any — its only purpose is to unlock subsequent simplifications that would otherwise be impossible. A puzzle that requires Move III before Move II cannot be solved by trying Move II first. The correct sequence must be found. Five levels introduce these ideas in sequence. The first four are projections of the unknot — a simple loop that looks tangled but can be completely simplified to zero crossings. The fifth is a trefoil. The trefoil has three crossings at minimum, no matter how it is drawn or what sequence of Reidemeister moves is applied. It is a genuine knot. Move I does not apply. Move II does not apply. The player attempts both and discovers that neither works, then identifies the knot by name. Three of its crossings light up in red, blue, and green — the colours of a valid tricolouring, a mathematical proof that the diagram cannot be the unknot. The game teaches without stating: that a messy-looking diagram can be simpler than a clean-looking one. That the crossing count in a diagram is not the crossing number of the knot — the trefoil always has at least three, but a projection of the unknot can have twenty. That some tangles come undone and some do not. That the difference is topological, not visual.