New examples of tunnel number subadditivity

If the tunnel number of a knot K is denoted t ( K ) , a pair of knots K 1 , K 2 is said to be subadditive if t ( K 1 ) + t ( K 2 ) > t ( K 1 # K 2 ) . Scharlemann and Schultens (2000) [11] defined the degeneration ratio to be d ( K 1 , K 2 ) = 1 − t ( K 1 # K 2 ) t ( K 1 ) + t ( K 2 ) , and proved that d ( K 1 , K 2 ) ⩽ 3 / 5 . However, the highest known degeneration ratio known for a pair of knots is just 2/5. We use free decompositions to construct links which experience degeneration approaching 3/7 when the connect sum is taken with certain knots. These links can be modified to yield a family of knots whose members we conjecture to have the same property.