Admissible transverse surgery does not preserve tightness

Abstract

We produce the first examples of closed, tight contact 3-manifolds which become overtwisted after performing admissible transverse surgeries. Along the way, we clarify the relationship between admissible transverse surgery and Legendrian surgery. We use this clarification to study a new invariant of transverse knots -namely, the range of slopes on which admissible transverse surgery preserves tightness -and to provide some new examples of knot types which are not uniformly thick. Our examples also illuminate several interesting new phenomena, including the existence of hyperbolic, universally tight contact 3-manifolds whose Heegaard Floer contact invariants vanish (and which are not weakly fillable); and the existence of open books with arbitrarily high fractional Dehn twist coefficients whose compatible contact structures are not deformations of co-orientable taut foliations.

Braids with as many full twists as strands realize the braid index

“…Since adding and removing generators yield a cobordism consisting of a 1-handle between the corresponding closures, this implies that there exists a cobordism of Euler characteristic −(n − 1)(nk + 1) + wr(α) between K and T n,nk+1 . Thus, we find t I (n − 1)nk 2 − I(K) = I(T n,nk+1 ) − I(K) (4) t I (n − 1)(nk + 1) − wr(α) 2…”

“…from which the result follows by taking the limit k → ∞. For instance, the 4-braid A = a 1 a 2 a 3 a 3 has fractional Dehn twist coefficient 1 3 (since one can first see using braid relations that A 3 = Δ 2 , and then apply (b) and (c) from Proposition 2.1), and so Υ A (t) has slope change 4 3 at 1 2 . The 5-braid B = a 1 a 2 a 3 a 4 a 1 a 2 also has fractional Dehn twist coefficient 1 3 and so Υ B (t) has slope change 5 3 at 2 5 .…”

“…The fractional Dehn twist coefficient has been extensively studied in the context of contact topology and open book decompositions (see, for instance, [4,[22][23][24][25]30]) and of course in the context of classical braids [35,36]. Relationships have also been explored between the fractional Dehn twist coefficient and monoids in the mapping class group [13,26], classical knot theory [30], and homological invariants of knots and 3-manifolds [22,42].…”

Symplectic Fillings, Contact Surgeries, and Lagrangian Disks

“…Instead of doing admissible transverse surgery on T , Baldwin and Etnyre showed in [1] that for certain s, the same result can achieved via Legendrian surgery. Theorem 2.4 (Baldwin-Etnyre, [1]). Let T ⊂ N come from contact (r)-surgery on L, with notation as above.…”

“…As an auxiliary tool in our study of surgeries on Legendrian knots we need to consider surgeries on transverse knots. We briefly recall admissible transverse surgery here, see [1] for more details.…”