In many idols of lord Ganesha, I saw that the left tusk is broken.
What is the story behind that broken tusk? Does it signify something?
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The story of how Ganesha broke his tusk is recounted in this excerpt from the Upodghata Pada of the Brahmanda Purana. Parashurama, the axe-wielding incarnation of Vishnu, had sucessfully defeated his enemy Kartavirya Arjuna and the kings allied with him, and so he wanted to thank Shiva for giving him the power to fight these enemies. Parashurama went to Mount Kailash to pay his obeisances to Shiva, but Ganesha stopped him, saying his father was sleeping along with his mother, and he didn't want Parashurama intruding on them in case they might be engaged in amorous pursuits. Parashurama was enraged that he was being prevented from seeing Shiva, and so he started fighting Ganesha. Ganesha was winning handily, but then Parashurama threw his axe at Ganesha and Ganesha didn't fight back against it, because the axe was a gift from Shiva:
Perceiving that the axe had been given to him by his father, Ganesha became desirous of meaning it not to go in vain. Hence he received it with his left tooth (tusk). Chopped off by the axe, the tusk fell on the ground, covered with blood like a mountain that fell on the ground when struck by Indra's thunderbolt.
Note that there are other stories about Ganesha losing his tusk, most famously the story included in some manuscripts of the Mahabharata concerning Ganesha breaking his own tusk off to continue writing the Mahabharata as Vyasa was dictating it to him. But this story is widely agreed by scholars to be a later interpolation, and it's not found in any critical editions of the Mahabharata, and you can see for yourself that it's not even included in this translation of chapter 1 of the Adi Parva. So the most prominent story that's actually grounded in scripture is the one I gave above.
Skanda Purana narrates different story for his broken tusk. It says Lord Ganesha broke his own tusk for YOGADANDA.
He (Ganesha) pulled out his own huge tooth for sake of "Yogadanda" (the mystic wand of Yogic practice) and held it in his hand.