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INVOLUTIONS OF KNOTS THAT FIX UNKNOTTING TUNNELS
Preview AbstractLet K be a knot that has an unknotting tunnel τ. We prove that K admits a strong involution that fixes τ pointwise if and only if K is a two-bridge knot and τ its upper or lower tunnel.
TWISTED TORUS KNOTS T(p, q; 3, s) ARE TUNNEL NUMBER ONE
Preview AbstractWe show that twisted torus knots T(p, q; 3, s) are tunnel number one. A short spanning arc connecting two adjacent twisted strands is an unknotting tunnel.
A CONSTRUCTION OF ARBITRARILY HIGH DEGENERATION OF TUNNEL NUMBERS OF KNOTS UNDER CONNECTED SUM
Preview AbstractWe show that the degenerations of tunnel numbers of knots under connected sum can be arbitrarily large.