Transverse Surgery on Knots in Contact 3-Manifolds

Abstract

We study the effect of surgery on transverse knots in contact 3-manifolds. In particular, we investigate the effect of such surgery on open books, the Heegaard Floer contact invariant, and tightness. One main aim of this paper is to show that in many contexts, transverse surgery is a more natural tool than surgery on Legendrian knots. We reinterpret contact <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis plus-or-minus 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mo> ± </mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\pm 1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -surgery on Legendrian knots as transverse surgery on transverse push-offs, allowing us to give simpler proofs of known results. We give the first result on the tightness of inadmissible transverse surgery (cf. contact <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis plus 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(+1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -surgery) for contact manifolds with vanishing Heegaard Floer contact invariant. In particular, inadmissible transverse <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r"> <mml:semantics> <mml:mi>r</mml:mi> <mml:annotation encoding="application/x-tex">r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -surgery on the connected binding of a genus <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g"> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding="application/x-tex">g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> open book that supports a tight contact structure preserves tightness if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r greater-than 2 g minus 1"> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>2</mml:mn> <mml:mi>g</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">r &gt; 2g-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .