PPERIODIC SPANNING SURFACES OF PERIODIC KNOTS
STANISLAV JABUKA
Abstract.
Edmonds [1] famoulsy proved that every periodic knot of genus g pos-sesses an equivariant Seifert surface of genus g . We show that this is not true if oneinstead considers nonorientable spanning surfaces of a periodic knot. We demonstrateby example that the difference between the first Betti number of an equivariant and anonequivariant nonorientable spanning surface of a periodic knot, can be arbitrarilylarge. A knot K in S is said to be periodic if there exists an integer p ≥
2, a diffeomorphism f : S → S of order p that preserves the knot K , and whose fixed point set Fix( f ) isdiffeomorphic to S . In this case we say that K is p -periodic, that p is a period of K ,and we call Fix( f ) the axis of f . See [3] for more background on periodic knots.In [1] Edmods famously proved that if K is a p -periodic knot of genus g , then thereexists a Seifert surface Σ for K of genus g that is invariant under the diffeomorphism f . Said differently, if we define the p -periodic (or equivariant ) g ,p ( K ) of a p -periodic knot K as g ,p ( K ) = min { g ≥ | K possesses an f -invariant Seifert surface of genus g } , then Edmonds’ theorem can be seen as saying that g ( K ) = g ,p ( K ) for every p -periodicknot K (with g ( K ) being the Seifert genus of K ).The goal of this note is to show that if one considers nonorientable spanning surfacesfor periodic knots instead, the analogue of Edmonds’ theorem is not true. To state ourresult, we recall the definition of the nonorientable (nonequivariant) 3-genus γ ( K ), andwe define the p -periodic (or equivariant ) nonorientable 3-genus γ ,p ( K ) of a p -periodicknot K as γ ( K ) = min { b (Σ) | Σ is a nonorienatble spanning surface for K } ,γ ,p ( K ) = min { b (Σ) | Σ is an f -invariant nonorienatble spanning surface for K } . We leave it as an easy exercise to show that every periodic knot has an equivariantnonorientable spanning surface, and thus the definition of γ ,p ( K ) is well posed. It isalso not hard to show that γ ,p ( K ) ≤ g ( K ) + p for any p -periodic knot K . Theorem 1.
Let K be a p -periodic knot with p ≥ and with γ ( K ) ≥ . Then γ ,p ( K ) ≥ p .Proof. Let f : S → S be a diffeomorphism that facilitates the p -periodicity of K andlet A =Fix( f ) be its axis. Let further Σ ⊂ S be a nonorientable f -invariant spanningsurface for K and let Σ ⊂ S be the quotient of Σ by the action of Z p generated by The author was partially supported by the Simons Foundation, Award ID 524394, and by the NSF,Grant No. DMS–1906413. a r X i v : . [ m a t h . G T ] J a n STANISLAV JABUKA
Figure 1.
The torus knot T (5 ,
3) shown with an equivariant nonori-entable spanning surface Σ with b (Σ) = 5. f , and note that Σ is nonorientable. Then Σ → Σ is a p -fold cyclic cover, branchedalong λ ≥ λ being the number of points in Σ ∩ A . A straightforwardcomputation of Euler characteristics gives(1) χ (Σ) = p · χ (Σ) − ( p − λ. Write b (Σ) = a and b (Σ) = b . The assumption γ ( K ) ≥ a ≥
2, while bydefinition b ≥ λ ≥
0. Equation (1) then becomes(2) a − p ( b −
1) + ( p − λ. If b = 1, we obtain a − p − λ forcing λ > a ≥
2. This in turn forcesthe inequality a − ≥ p − a ≥ p . If b ≥ a − ≥ p . Thus, ineither case we find a ≥ p and hence γ ,p ( K p ) ≥ p , since Σ was an arbitrary equivariantnonorientable spanning surface for K . (cid:3) Corollary 2.
The difference between the equivariant and nonequivariant nonorientable3-genera of a periodic knot can become arbitrarily large. Specifically, for every integer p ≥ there exists a p -periodic knot K p with γ ( K p ) = 2 and γ ,p ( K p ) ≥ p. Proof.
Let K p be the torus knot T (4 p, p − γ ( K p ) = 2for all p ≥
3. The periods of a torus knot T ( a, b ) are precisely the divisors of | a | and | b | , showing that K p is p -periodic. Theorem 1 implies that γ ,p ( K p ) ≥ p . (cid:3) The inequality γ ,p ( K ) ≥ p from Theorem 1 is sharp as seen in the next example. Example . Consider the 5-periodic torus knot K = T (5 , γ ( K ) = 2 (or use [5] where T (5 ,
3) is the knot 10 ), showing that K meets thehypothesis of Theorem 1 and thus γ , ( K ) ≥
5. An equivariant spanning surface Σ for K with b (Σ) = 5 is shown in Figure 1, leading to γ , ( K ) = 5. The values of a , b , λ from the proof of Theorem 1 are 5, 1, 1 respectively, and satisfy equation (2).Another important result of Edmonds’ [1] is the bound p ≤ g ( K ) + 1 satisfied byany period p of the knot K . While it was known prior to Edmonds’ work that a knot ERIODIC SPANNING SURFACES OF PERIODIC KNOTS 3 may only have finitely many periods (cf. Theorem 3 in [2]), the preceding inequality wasthe first quantitative bound on the number of possible periods of a knot. Corollary2 shows, as yet another contrast to Edmonds’ results, that no upper bound on theperiods of a knot can exist by any polynomial function in the nonorientable 3-genus.This conclusion also follows from considering the p -periodic alternating torus knots T (2 , p ) for which γ ( T (2 , p )) = 1 = γ ,p ( T (2 , p )), with p ≥ References [1] Allan L. Edmonds. Least area Seifert surfaces and periodic knots.
Topology Appl. , 18(2-3):109–113,1984.[2] Erica Flapan. Infinitely periodic knots.
Canad. J. Math. , 37(1):17–28, 1985.[3] Stanislav Jabuka and Swatee Naik. Periodic knots and Heegaard Floer correction terms.
J. Eur.Math. Soc. (JEMS) , 18(8):1651–1674, 2016.[4] Stanislav Jabuka and Cornelia A. Van Cott. Comparing nonorientable three genus and nonori-entable four genus of torus knots.
J. Knot Theory Ramifications , 29(3):2050013, 15, 2020.[5] Charles Livingston and Allison H. Moore. Knotinfo: Table of knot invariants. URL: knotinfo.math.indiana.edu , Current Month Current Year.[6] Masakazu Teragaito. Crosscap numbers of torus knots.
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Email address : [email protected]@unr.edu