How does the knot orientation interact with other properties or invariants.

Question

The knot table consists of prime, non-oriented knots, not including the mirror images. However, for signature to be defined, we have to assign orientation to knots. How does the orientation influence the knot table and chirality? Do we simply obtain 4 knots instead of 1 in the table in case of chiral knots, and 2 instead of 1 in case of amphichiral? Is the signature the same for, say, left-handed trefoils oriented in "opposite" directions? Meticulous response would be appreciated.

Answer 1

I am going to quote Wolfram Mathworld, twice. First, for invertible:

An invertible knot is a knot that can be deformed via an ambient isotopy into itself but with the orientation reversed. A knot that is not invertible is said to be noninvertible.

Amphichiral:

An amphichiral knot is a knot that is capable of being continuously deformed into its own mirror image.

A knot is either invertible or not and either amphichiral or not, and these are not related. For convenience, lets say that $(i,a)$ is invertible and amphichiral. Then $(i,-a), (-i,a)$, and $(-i,-a)$ are the other three options. There are examples of each of these.

As for the knot table, like you stated, usually there is only one knot drawn, regardless of invertibility or chirality.

As to whether signature distinguishes knots and their inverses, think about how we define signature, and then consider what would happen if we switched the orientation. Good luck.