This is the 10982nd most frequent Russian word.
вязаном
"Вязаном" means "knitted" (describing something made by knitting).
The word 'вязаном' is used here as an adjective describing the style of the sweater, indicating that it is knitted.
Она носила свитер в вязаном стиле.
She wore a sweater in a knitted style.
Here, 'вязаном' qualifies 'носок', indicating that the sock has a knitted quality, emphasizing its cozy nature.
Вязаном носке удобно зимой.
A knitted sock is comfortable in winter.
In this context, 'вязаном' describes the small house for the cat, specifying that it is made from knitted materials.