What is known about the Kaustubh Mani that came out of the Samudra Manthana?
This wiki article says that the gem is worn by Vishnu.
Are there any texts which go into more details?
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The Word "Kausthubha" is a combination of two words,
Kausa+Abha(कौस्थ+अभा).Abha means Light or Grace , Luster , Brightness etc.
सशंख चक्रं सकिरीत कुण्डलं, सपीत वस्त्रं सर्सि रुहेक्षनं ।
सहार वक्षस्थल कौस्तुभश्रियं, नमामि विष्णुं शिरसा चतुर्भुजं ।।
sashankha chakraM sakiriita kuNDalaM, sapiita vastraM sarsi ruhekshanaM sahaar vakshasthala kaustubhashriyaM, namaami viShNuM shirsaa chaturbhujaM Vishnu Stuti
We find mentioning of Kausthubha Mani in various places in puranas.
कौस्तुभाख्यमभूद् रत्नं पद्मरागो महोदधे:।
तस्मिन्हरी: स्पृहं चक्रे वक्षोsलङ्करणे मणौ॥५॥
kaustubhākhyam abhūd ratnaṁ padmarāgo mahodadheḥ tasmin maṇau spṛhāṁ cakre vakṣo-’laṅkaraṇe hariḥ
Meaning- Generated thereafter from the great ocean were the celebrated gems Kaustubha-maṇi and Padmarāga-maṇi. Lord Viṣṇu, to decorate His chest, desired to possess them. Generated next was the pārijāta flower, which decorates the celestial planets. O King, as you fulfill the desires of everyone on this planet by fulfilling all ambitions, the pārijāta fulfills the desires of everyone.Shreemad Bhagvatam
According to various legends -:
Kausthubha Mani (Gem , Jewel) (कौस्तुभ मणी) is a very precious jewel came out "Samudra Manthan" or The churning of the Ocean of milk by Devatas & Asuras. This Jewel is worn by "Lord Vishnu" (भगवान विष्णू) On his chest.
"Kausthubha Mani" is very bright jewel This "Kausthubha Mani" is said to be having no hole or defect in it. And also no string attached to it Because "lord Vishnu" , does not like any defect or string ( String word which is used here as to denote non bondage to anything).
It is believed that where this Kausthubha Mani rests , no calamity occures there. In another belief this jewel is representation of Lord Vishnu's brightness as Sun.