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I am computing force between two magnetic poles each of one unit pole (in emu) and situated one centimeter apart.

In electromagnetic units: $$F_{dyne}=\dfrac{p^2}{r_{cm}^2}=\dfrac{1^2}{1^2}=1 dyne$$ where $p$ is pole strength in emu

In SI units: $$F_{N}=k_A \dfrac{P^2}{r_m^2}=10^{-7} \dfrac{({1.25\times 10^{-7}})^2}{10^{-4}}=1.5625 \times 10^{-17} \neq 10^{-5}N=1dyne$$ where $P$ is that same pole strength in SI units

with $P=1.25\times10^{-7}p$ see here

Now why am I getting two different results in emu and SI for the same configuration?

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I find this topic to be a quagmire.

The SI unit of magnetic pole strength is the ampere-metre with $1\,\rm Am$ equal to $10$ electromagnetic units (emu) of magnetic pole strength.

The relationship is derived here.

The conversion that you were using was for magnetic flux the SI unit of which is the weber and it is the maxwell in emu with $1$ maxwell being equal to one line of force and a unit magnetic pole produces $4\pi$ lines of force.
You will note that the site that you quoted for your conversion gives this relationship between magnetic pole strength and the flux produced in maxwells.

In terms of magnetic flux $1\,\rm weber = 10^8\, \rm maxwell$
Now a unit pole produces $4\pi \,\rm maxwell$ which is $\dfrac {4\pi}{10^8} = 1.2 \times 10^{-7} \,\rm Wb$

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