Convicted double murderer Mark Lundy is pleading for a final chance to appeal in court over inadmissible scientific evidence and his much-publicised funeral behaviour, both of which he believes were unfairly allowed to skew his second trial.

Lundy's first conviction for the 2000 murder of his wife Christine and daughter Amber, 7 - after which an appeal led to his prison sentence being increased - was quashed by the Privy Council in 2013.

At his second trial, in 2015, he was again convicted. The Court of Appeal rejected his appeal last year and Lundy has asked the Supreme Court for its permission to be heard there.

Images of Mark Lundy being supported after the funeral of his wife Christine and daughter Amber in 2000 were
Images of Mark Lundy being supported after the funeral of his wife Christine and daughter Amber in 2000 were "folklore in New Zealand", his legal team says. Photo / Herald file

The murder weapon with which Christine and Amber were hacked to death in the family home at Palmerston North wasn't found, but was likely to have been an axe or tomahawk.

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Scientific evidence about allegedly human material on Lundy's shirt is contested.

Lundy's defence has argued that the 2015 jury should not have heard the evidence, based on micro-RNA analysis, as the technique was too new and unproven to be relied on.

In a notice to the Supreme Court outlining the case for the new appeal, Lundy's legal team, headed by Jonathan Eaton QC, said the Court of Appeal was wrong to conclude the second trial was fair despite the admission of "inadmissible" micro-RNA scientific evidence.

In the notice, Lundy says the Court of Appeal dismissed his appeal despite ruling the evidence of a crown witness in relation to the alleged presence of micro-RNA material to be scientifically invalid and inadmissible.

Eaton argues in the notice that the Court of Appeal's errors included concluding that the m-RNA evidence was not significant enough in the overall context of the crown's case for its wrongful admission to have rendered the trial unfair.

The appeal judges failed to account for the "illegitimate and unfair bolstering effect" the inadmissible evidence had on the crown's admissible evidence.

Eaton also notes the effect of the "infamous Lundy funeral scene".

Images were published of Lundy staggering and being supported by the arms at the funeral of Christine and Amber.

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Eaton's notice says the scene had "become folklore in New Zealand, was referenced in the defence opening statement and was the elephant in the room at Lundy's second trial.

"It needed to be addressed by specific [judicial] direction in order to avoid the risk of the jury engaging in demeanour reasoning."