is there a synonym for "manipulation" that carries the connotation of "influence by artful means" but lacks the connotation of "physical interaction involving hands"?
I am going to add puppet master because puppet master controls from behind the scenes, manipulating people and things like a puppeteer operating a puppet. http://www.dictionary.com/e/slang/puppet-master/
n. sway TFD
n. 2. Influence or control:
I was going to provide coercion, but that implies intimidation. The question specifically made use of the term "artful means"—which is something more subtle and intellectual.
to obtain by indirect or involved means · finagle a ride home
to obtain by trickery · finagled his way into the concert
to use devious or dishonest methods to achieve one's ends · A con man finagled my neighbor out of $400.
let me look at my schedule and see if I can't finagle a visit to the museum
Edited to add an alternative in the same vein.
a person who supplies the directing or creative intelligence for a project
a person who plans and organizes something
the real mastermind behind the embezzlement scheme
They masterminded a unique solution to the problem.
Here is one last attempt, in a different sense. If this is also not what you're after, you will need to provide some specific sentence examples with "blanks" so that the meaning you're after is clearer.
1 : hoodwink · beguiled her classmates into doing the work for her
2 : to engage the interest of by or as if by guile · His seductive voice beguiled the audience.
3 : to lead by deception · beguiled into ambush
to deceive by wiles · had intended to beguile
Something can also be beguiling:
agreeably or charmingly attractive or pleasing · a beguiling manner · a beguiling aroma
Handling: the action of touching with the hands (or the skillful use of the hands) or by the use of mechanical means