Feminine nouns ending in -а or -я lose that ending in the genitive plural. Frequently there is an extra -e- or -o- shoved inbetween the last consonants to facilitate pronunciation. The ending for the genitive plural is called a "zero-morpheme" (like: кошк-ø, which then surfaces as кошек). The genitive plural of книга, for instance, is книг.

"кошки" (plural) becomes "кошек" because "нет" takes genitive.

беглые гласные ‧ Escaped / Runaway / Fugitive Vowels
Russian Fleeting / Fill / Inserted / Mobile Vowels
‧ ‧ Fleeting Vowels ‧

[ окно окон ] ‧ [отец отцу] ‧ [ ружьё ружий ] ‧ fleeting vowel ‧ inserted vowel ‧ examples: палец, рынок, ужасен, лёд (compare to льдом), свадеб (compare to свадьба), чаек (compare to чайка), заём (compare to займа) ‧ Cornell U Ru ‧

Genitive plural, nouns with a stem ending in two (or more) consonants need a vowel (o or e) which is inserted before the last consonant. In most cases, e is inserted if the stem end in ж/ш/ч/ц/щ/soft с + consonant or a soft consonant + consonant, and о is used if the stem ends in another combination of consonants. ‧ Ru genitive plural ‧ fleeting vowel

‧ ле́в => львы́ ‧ Mobile Vowel ‧ отец => отцы ‧ ‧ ле́в declension

because that is the plural way how you say it.

That is exactly my question.

my guess is "у меня нет кошки".