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I know that trefoil knots do not have a mirror image, but I also know that there is a left-handed and a right-handed trefoil knot. I have the drawings of them. Isn't this a contradiction?

Rodrigo de Azevedo
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Emptymind
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    Max Dehn proved the right- and left-handed trefoils were distinct in 1914 https://math.stackexchange.com/questions/2511364/how-did-dehn-prove-that-the-trefoil-is-chiral – Kyle Miller Jul 03 '19 at 02:15

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What do you mean by „do not have mirror images“? The left handed tre foil knot is exactly the mirror image of the right handed one and vice versa. Therefore there is no contradiction. Actually, the tre foil knot is the easiest example of a chiral knot, a knot not being equivalent to its mirror image.

Con
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  • On the last line of page 9, and on page 10 in the book of Richard H. Crowell and Ralph H. Fox of knot theory they said that it is not amphicheiral and I understand that chiral and amphicheiral has the same meaning ..... am I correct? If no, what is the difference? – Emptymind Jul 04 '19 at 20:11
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    No, they are opposites. Chiral knots are knots that are not equivalent to their mirror image and amphichiral knots are knots are equivalent to their mirror image. Therefore not amphichiral is the same as chiral. – Con Jul 04 '19 at 20:49