Are 3#4 and 3*#4composite knots isotopic?

Question

all I found (on wolfram) that there is one composite knot with seven crossings and that is the 3#4. But is this really equivalent to 3*#4 i.e. a composite knot of trefoil with opposite chirality and prime knot with 4 crossings. ? Thanks!

Answer 1

The signature of a knot is additive under connected sum, changes sign when you do a reflection, and is $-2$ on the trefoil knot. So the signature of $3_1 \sharp 4_1$ is $-2$ and the signature of $\overline{3_1} \sharp 4_1$ is $2$. Therefore they are not isotopic.

Of course, since $\overline{4_1} = 4_1$, one is actually the mirror image of the other, which may explain why the two are not separately listed on Wolfram.