New examples of tunnel number subadditivity

If the tunnel number of a knot K is denoted t(K), a pair of knots K1,K2 is said to be subadditive if t(K1)+t(K2)>t(K1#K2). Scharlemann and Schultens (2000) [11] defined the degeneration ratio to be , and proved that d(K1,K2)⩽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.