What is the chemical formula(s) for liquid potassium-based soap?
You'll get an answer quickly if you search for &quot;formulation liquid soap&quot;. There are for instance DIY guides on making liquid soap starting from natural oils (sunflower, coconut) by saponifying the oil with KOH. The fatty acid composition of sunflower oil can vary significantly depending on breed (see e.g. the Wikipedia) but generally contains mono- and polyunsaturated FAs in large proportion. Unsaturation contributes to the fluidity of the product. It is also worthwhile to check common nomenclature for FA chain length (e.g. see the Wikipedia). Sunflower oil contains palmitic (C16:0) and stearic (C18:0) acids as principal saturated FAs. Coconut oil contains shorter (lauric, C12:0) and mainly saturated FAs. Mainly because of the extent of saturation, coconut oil has a higher melting point and is used as the minor component.
In addition there are plenty of commercial vendors willing to provide recipes you can follow to make your own liquid soap. Note not all require a potassium salt. For instance, the company Stepan provides a recipe which consists of mixing the following (in decreasing order of weight %) to form one phase (which is then mixed with another lacking detergents):

Deionized Water
BIO-TERGE® AS-40 (or BIO-TERGE® AS-40 HP)  ==  SODIUM C14-16 OLEFIN SULFONATE
STEPANOL® AM  == AMMONIUM LAURYL SULFATE
STEPANOL® WAT == TEA LAURYL SULFATE
NINOL® M10  == COCAMIDE MIPA

Admittedly ammonium counterion (and TEA) is like potassium in that they are larger and softer counterions than sodium, so similar in their effect on the detergent.