Well, yok means "there is not". Saying "Senin Sekerin yok" means "Your sugar doesn't exist" or "You don't have sugar" and is the opposite of var, which means "there is". To us, var may as well be interpreted as "have", and yok as "have not" in this possessive context.
According to another source, "yoktur" means "definitely non-existent", where as "vardir" means "definitely is". They are used to emphasize factual statements, which I believe was previously stated. My Turkish friends have never corrected my usage of "yok" though, so this is somewhat new to me.
Degil means "not". If you said.. "Senin Sekerein degil" that is negating the implication that the sugar was yours. So, it's like saying "(It's) not your sugar". Does this make sense?
Auxiliary translations like yours that mirror the gramatical structure mean a lot to me! They reflect that "other way" of putting things into words so that they become more comprehensible. Thank you for that bridge!
I would be delighted, if Duo would make use of this approach more often where everything is accessible for mobile devices - and not only in the ommentary section!
(For more inspiration about similar thoughts and tecniques see also the findings of -> Vera F. Birkenbiehl)