Un-knot 1 and 2
These diagrams follow all of the conventions of knot diagrams, although one does not exhibit the smooth lines that are normally seen (but not prescribed). Despite all their loops and crossings, because of the way they are drawn—the line always ‘passes behind’ any existing line, signified by a line break—these knots will always untangle unproblematically to become a loop, the un-knot, the least mathematically interesting configuration possible.
To draw a mathematically interesting, non-untangleable knot, the line has to be broken earlier in the sequence in expectation of future crossing lines, an uncomfortable drawing gesture that necessitates thinking ahead of the flow of the line.
Modelling from diagrams—incomplete three-dimensional models of the mental imagery of transitory diagramming procedures
Diagrams taken from Jessica S. Purcell, “Geometric Techniques in Knot Theory”, videosfromIAS YouTube channel. 2015