This book at page 212 says:
the lord in yoga nidra pose is offering bilva leaves to Lord Siva
So, is the Padmanabhaswamy idol at Trivandrum represented as worshipping Siva linga?
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No, its not worshipping the shiva lingam.
The official website of Ananta Padmanabhaswamy temple has a different story to tell. It says that the Shiva Linga's presence in the idol might have been linked to the killing of the demon Keshi or the story of Bhasmasura. (Sri Anantha Padmanabha Swamy Temple)
The site doesn't mention the relation between the demon Keshi and Shiva. All we know is that the word Keshi means the hairy one, which is affixed in front of the first teerthankar of Jains, Rishabha and hence the name Keshi Rishabh. Keshi Rishabha has similar similarities with that of Lord Shiva (to know more check Shiv and Rishabh : A study), and the word Keshi can also mean Jatadhari, another attribute of Shiva. But since the site has affixed Demon, it might be mentioning the demon Keshi, the hairy horse, sent by Kamsa to kill Krishna, the eighth Avatara of Shri Hari Vishnu, who was however defeated, in the Dwapara Yuga.
The next story is related to Bhasmasura. Lord Shiva always smears Bhasma or ash on his body. He is also Bholenath, the innocent God. When a demon worshipped him very piously, Shiva appeared before him and wished him to grant a boon. The demon replied that he wanted to spend his life in Seva of Shiva ie working for him. Shiva agreed and asked him to collect the ash from the earth and bring it to him. The demon continued to do his work but he felt it was very tiresome so he asked Shiva for another boon, that whatever he touches shall be transformed to ash. Shiva granted him so. Infact the demon had seen Parvati and wanted to posses her which couldn't have been possible in the presence of Shiva. So he moves to Shiva himself to test his boon and also reduce Shiva to ashes. Shiva flees to Vishnu and asks for his protection. Vishnu readily agrees to this and asks Shiva to hide behind a tree. Vishnu then takes his most beautiful Mohini form and appears before the Bhasmasura (the demon/Asura of Bhasma or ash). Seeing Mohini, Bhasma forgets about Parvati and asks for her hand. Mohini agrees to wed Bhasmasura on only one condition, if he is able to match her steps as she dances. Bhasmasura agrees to her and starts imitating the exact steps Mohini was performing. At last Mohini placed her right hand on her head and when Bhasmasura did the same he himself was reduced to ashes. Thus Vishnu had protected Shiva. In the idol, Vishnu places his hand over Shiva in Abhaya Mudra, asking him not to fear.
Now there is another perception to placement of Shiva/ Shiva Linga under Vishnu's right arm.
Read more at official site- http://www.ananthapadmanabhaswamytemple.org/deities.html
It is a representation of cosmic process.
Touching shivaling is the repersentation of dissolution of creation.
Placing hand on shiva linga shows that he is above dissolution of world death and recreation, he is supreme. Here shiva lingam is representation of dissolution.
The mohini-shiva story- http://hindumythologyforgennext.blogspot.in/2012/04/bhasmasura-and-mohini.html?m=1
The ayyapan story-harihara story were very famous in south india- padmanabhaswamy is also shows the mixing of shiva and Vishnu worship. Shiva is incomplete without Vishnu and Vishnu without shiva.
Also Lord Vishnu has a flower in one hand but he doesn't have any bilva in other hand.