What would happen if two knots were merged together, like the two below? Could the combination simplify more than either of the original knots? Or could they even cancel each other out, leaving only the unknot? What does this do to tricolorability? What would this do to the polynomial invariants (Jones, HOMFLY, Alexander)? Can a knot be "factored" into two or more knots like the one above? Can any knot be factored?
These are all common questions dealing with knot arithmetic. Knot arithmetic is the branch of knot theory involving the properties of knots when they are combined or seperated.
Curiously, most properties of a combined knot can be derived from the corresponding properties of its components, like general arithmetic and natural numbers. As in natural numbers, there is no multiplicative inverse in knot arithmetic, so knots can not cancel each other out. When knots are combined, the genus and bridge number behave like addition. The polynomial invariants, on the other hand, generally behave like multiplication. There are prime knots and composite knots.Previous Next