Left Orderability of Dehn Surgery on Odd Classical Pretzel Knots
Khan, Md Arafat Hossain
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A non-trivial group G is called left-orderable if there exists a strict total ordering < on its elements such that g < h ⇒ fg < fh for all f, g, h ∈ G. Wallace and Lickorish independently proved that every closed, orientable, connected 3-manifold M is obtained by performing Dehn surgery on a framed link in S³ with ±1 surgery coefficients [76, 45]. All knot groups are left-orderable . But the fundamental group of the manifold obtained by Dehn surgery on a link/knot is not necessarily left-orderable. As an example, many knots admit Dehn surgery yielding lens spaces with non-left-orderable fundamental groups. An L-space is a rational homology sphere Y whose Heegaard-Floer homology group HF(Y) has rank equal to |H₁(Y ; ℤ)| . Boyer, Gordon and Watson conjectured that an irreducible rational homology 3-sphere is an L-space if and only if its fundamental group is not left-orderable . The conjecture was confirmed for all graph manifolds , many branched covers of knots in the 3-sphere , more than 100, 000 small-volume hyperbolic 3-manifolds , Seifert fibered manifolds, Sol manifolds, double branched covers of non-splitting alternating links  etc. Hakamata and Teragaito showed that if 0 ≤ r ≤ 4 then rsurgery on any hyperbolic twist knot yields a manifold whose fundamental group is left-orderable . Tran extended this result for twist knots with m crossings to the interval (−4, 2m) for even m and to the interval [0, 4] ∪ (4m w + 4, 2m + 4) for odd m where, w > 1 is the unique real solution of the equation xex = 4m . Lidman and Moore proved that pretzel knots admit an Lspace surgery if and only if it is isotopic to ±(−2, 3, q) for odd q ≥ 1 or torus knots T(2, 2n + 1) for some n . As a corollary we obtain that no odd classical pretzel knots (see Figure 3.2) admit an L-space surgery. More generally, Ozsvath and Szab ´ o proved that alternating hyperbolic ´ knots do not admit L-space surgery. Odd Classical Pretzel knots are alternating and hyperbolic. The conjectured connection between L-spaces and left-orderability of their fundamental groups is the main motivation for this research to prove left-orderability of an infinite family of closed, orientable, connected 3-manifolds obtained by performing Dehn surgeries on odd classical pretzel knots. So, if we support the conjecture, we expect that manifolds obtained by any Dehn surgery on any odd classical pretzel knot must have left-orderable fundamental group. The main achievement of this research is to tackle knots whose crossing number varies with three different parameters. All the works done so far used at most two parameters to define the crossing numbers. In that sense this research is a door opening for more complicated knots. The closest paper that dealt with three parameter knot group was only varying the crossing number of one tangle , i.e. (−2, 3, 2s+1)- pretzel knots. In this research we let all the tangles vary, i.e. (2k₁+1, 2k₂+1, 2k₃+1)-pretzel knots and investigate the left orderability of those. We obtained a continuously varying one parameter family of elliptic representations and thereby we proved that all (2k₁ + 1, 2k₂ + 1, 2k₃ + 1)-pretzel knots for k₁, k₂, k₃ ∈ N admit left orderable fundamental groups obtained by r-Dehn surgery where r ∈ (−∞, 0) ∩ Q.