Salam, is the stitching on the socks of a woman which makes the skin colour visible make a person’s prayer and tawaf invalid? I tried to wear two layers of socks but the skin colour is still faintly visible.
Among the valid conditions of prayer is that the awrah must be covered. According to madhhab Syafie, the awrah of a man is a part of his body from the navel to the knees, whereas for a woman her whole body except her face and palms. If a man or woman performs prayer while not covering the awrah, then his/her prayer is invalid. Syafi’iyyah scholars supported this opinion with the hadith of Prophet PBUH narrated by Aisyah radhiyallahu ‘anha:
لَا يَقْبَلُ اللَّهُ صَلَاةَ حَائِضٍ إِلَّا بِخِمَارٍ
“The Salat (prayer) of a woman, who has reached puberty, is not accepted unless she is wearing a Khimar.” 
Imam al-Nawawi said, “what is meant by a woman who has reached puberty is a woman who has menstruated.” And this is also a valid condition for man.  It means Allah doesn’t accept the prayer of a man or woman who has reached puberty except when both has covered their awrah.
Imam al-Syairazi stated, the awrah is considered to be covered when the skin colour is no longer visible.  Syeikh Zainuddin al-Malibari explained that skin colour cannot be visible at a talking distance [the usual distance for two or people talking to each other] . In fiqh, the term used to describe this is majlis al-takhatub (discussion/chatting). Syeikh Habib Hasan bin Ahmad al-Kaf explained the distance of majlis al-takhatub (in the chapter on washing a thick beard) is approximately 1.5 metres.  Syeikh Nawawi al-Bantani stated: “[covering awrah] with a pure matter that could hinder the visibility of another of the skin colour, such that the fairness (of the skin) or darkness in a majlis al-takhatub.” 
Do the scholars analogically deduce the chapter of ablution with the chapter of awrah? Yes, Imam Ahmad bin Musa bin ‘Ajil, as stated by Syeikh Zainuddin al-Malibari  and Syeikh al-Syirwani.  However, according to Syeikh Abu Makhramah the final opinion states there is no difference between majlis al-takhatub or not. If a person has to look at the article of clothing carefully closely, then it is fine even if the awrah is still visible. 
To conclude, thin women’s socks that when worn the skin colour isn’t visible at a distance of 1.5 metres is permissible to be worn while praying. However, it is sunnah to wear a thicker pair of socks or wear longer praying clothes to avoid the khilaf. Wallahu a’lam.