Irena Turnau's work is somewhat problematic and lacking in references, and there are also a few places where she seems to get her facts mixed. She developed a theory about the origins of knitting that as far as I know is shared by no one else, and seems to be contradicted by some of the early evidence -- partly, I think, because she was working in almost complete isolation from other knitting historians, and also because the structure of knitted fabric was not really her main concern: she was far more interested in the economic and social aspects of home-made versus commercial textiles, so for her purposes the details of technique were far less important.
The English translation of her book also suffers from many serious typographical errors (and possibly translation problems as well -- I don't read Polish so I have no way of knowing).
For the early Islamic pieces, the best resource I've found is:
ANDRÉ, PAUL: Tissus d’Égypte: Témoins du monde arabe, VIIIe-XVe siècles. Éditions de l’Albaron, Société Présence du Livre, 1993,ISBN 2-908-528-525.
This book (in French) presents the Bouvier Collection of Arabic textiles: woven, embroidered, printed, and including fifteen fragments of knitting. Most have color photos. This is more pieces of early knitting in one place than I've seen anywhere else, even though they're only a small part of the over 200 textiles in the book. Unfortunately this private collection does not seem to be available to researchers. And like most private collections, the textiles in it were purchased from antiquities dealers, which means they have little or no archaeological context.
To the best of my knowledge, none of the early Islamic pieces have yet been carbon-dated. I believe they have mostly been dated by comparing the decorative motifs used to those in other more securely dated textiles.
The earliest pieces in this book may be from as early as the 9th century, but the 11th is perhaps more probable -- the evidence being motifs popular in that century.
The earliest date on a Spanish piece that I'm aware of is a glove from the tomb of Archbishop Rodrigo Ximenez de Rada, who died in 1245. The cushion from the tomb of Fernando de la Cerda, who died in 1275, seems to be the next oldest. I don't know whether any of the other 13th century fragments can be dated to a certain decade.
I can recommend two articles that might help.
Caune A., and Zarina A., 'Rigas 13.-15. gs. vilnas cimdi'. Latvijas PSR zinatnu akademijas vestis, 1:39, (1980). 60-9.
(If I remember correctly, the 13th century finds are nalbinding, but there is a knitted glove from the 15th century.)
Peets, J. 'Totenhandschuhe im Bestattungsbrauchtum der Esten und anderen Ostseefinnen.' Fennoscandia archaeologica IV, (1987). 105-116.
(Largely nalbinding again, but it also discusses that 13th century mitten fragment from Estonia.)
From attempting once to translate some of Turnau from the original Polish, the problem with the English stuff was (as best as my dictionary could help) that a lot of the textile terms were translated vaguely, because the original terms seem to be vague themselves, and not by someone knowledgeable in English textile terms.
A lot of things that might have been nalbinded are described as crocheted, for example, (but Turnau isn't the only scholar to do this, and the English summaries become more incomprehensible as terms are awkwardly butchered the older the article.)
Thanks for the references!
Regardless of its content, the paper by Turnau is just badly written (strange arguments, bad structure..). I don't understand how it could ever pass peer review... There are some interesting pictures in it though!
These are my favorite medieval knitting references:
Schmedding, B. (1978), Mittelalterliche Textilien in Kirchen und Klöstern der Schweiz, Bern: Abegg-Stiftung
-pictures and technical details of 14th c. knitted silk pouches
Gomez-Moreno, M (1946), El panteon real de las Huelgas de Burgos, Madrid: Consejo superior de investigaciones cientificas, Instituto Diego Velazquez
- pictures and technical data of 13th century knitted silk pillows