Calculating 4-cycles from $S_4$

Question

I'm learning about labeling knots with permutations from $S_n$ and I'm having trouble calculating the 4-cycles. I know, from an example in the book that
$$(1 2 3 4)(1 2 4 3)(1 2 3 4)^{-1} = (1 3 2 4)$$ so
$$(1 2 3 4)(1 2 4 3)(4 3 2 1) = (1 3 2 4)$$
And I just cannot for the life of me figure out how. If someone could break it down and explain the process, that would be wonderful.

Answer 1

You just have to use the formula for the conjugate of a cycle by a permutation $\sigma$: $$\sigma(1\,2\,4\,3)\sigma^{-1}=\bigl(\sigma(1)\,\sigma(2)\,\sigma(4)\,\sigma(3)\bigr)=(2\,3\,1\,4)=(1\,4\,2\,3).$$

Justification:

$$\sigma(1)\mapsto \sigma^{-1}\sigma(1)=1\mapsto2\mapsto\sigma(2).$$