There are knots whose tunnel numbers go down under connected sum

Abstract

In this paper, we show that there are infinitely many tunnel number two knots <italic>K</italic> such that the tunnel number of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K number-sign upper K prime"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mi mathvariant="normal">#</mml:mi> <mml:msup> <mml:mi>K</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">K\# K’</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is equal to two again for any 2-bridge knot <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K prime"> <mml:semantics> <mml:msup> <mml:mi>K</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">K’</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.