Lets say I have an oriented positive link L (i.e. an oriented link which admits a diagram with only positive crossings) such that L=L1#L2 for two links L1, L2.
Now my question is wether those two links L1 and L2 must also be positive.
I checked many low crossing examples with SnapPy of low crossing knots from knotinfo but have found no counterexample to this so far. My proofidea was to start with a positive diagram of L and try to reduce the L1 link in L so that it only intersects with itself such that I do not create negative crossings, but since the link L1 can seem heavily entangled inside the positive diagram of L, I am not sure how to this. I cannot seem to find an answer to this question even for knots. I know that if one replaces positivity with fiberedness, the answer is yes.
I also have the same question but for adequacy of a link instead of positivity (A link is called 0-adequate if it admits an 0-adequate diagram, i.e. a diagram such that doing 0-resolutions at all crossings and creating a graph whose vertices correspond to the resulting circles and the edges correspond to the crossings results in a graph (called seifert graph) without loops. Analogously we call a link 1-adequate by doing 1-resolutions instead of 0-resolutions. A link is adequate if it is both 0 and 1-adequate. Here is a link to a picture of what resolutions look like: https://miro.medium.com/max/640/0*ZpMXNFvIIvXYYFgz)
Thanks for any help.