It depends. The partnership would have filed the LLET (which is the Limited Liability Entity Tax, for LLC registration in KY). You the individual are not responsible for this filing. However, you are responsible for KY state tax on income generated in Kentucky. Although there is a reciprocal agreement between Kentucky and Michigan, the reciprocal agreement covers wages, salaries and tips earned for services rendered, whereas your income is a form of self-employment. Therefore, the net amount that you earn in Kentucky is taxable to Kentucky, requiring you to file a KY nonresident return.
Having said this, if you do need to file the KY return (you are required to if you have positive income in KY), then you will get a credit on your Michigan return for taxes you pay to KY on this income. This income is not taxable to Arkansas because you are a nonresident in Arkansas as well and this income was earned in Kentucky.
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