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If A+B=0, $A+B=0$, then A $$A = A + (B+A)+(B+A)+..=(A+B)+(A+B)+..=0.B+A)+(B+A)+\cdots=(A+B)+(A+B)+\cdots=0.$$
This is like the traditional joke proof that 1=0 $1=0$ with A=1, B=-1; $A=1$, $B=-1$; the difference is that the proof with knots is valid because the infinite sums of knots are meaningful: make the knots smaller and smaller.