Cosmetic crossing conjecture for genus one knots with non-trivial Alexander polynomial
aa r X i v : . [ m a t h . G T ] F e b COSMETIC CROSSING CONJECTURE FOR GENUS ONEKNOTS WITH NON-TRIVIAL ALEXANDER POLYNOMIAL
TETSUYA ITO
Abstract.
We prove the cosmetic crossing conjecture for genus one knots withnon-trivial Alexander polynomial. We also prove the conjecture for genus oneknots with trivial Alexander polynomial, under some additional assumptions. Introduction A cosmetic crossing is a non-nugatory crossing such that the crossing change atthe crossing preserves the knot. A cosmetic crossing conjecture [Kir, Problem 1.58]asserts there are no such crossings. Conjecture 1 (Cosmetic crossing conjecture) . An oriented knot K in S does nothave cosmetic crossings. Here a crossing c of a knot diagram D is nugatory if there is a circle C on theprojection plane that transverse to the diagram D only at c . Obviously the crossingchange at a nugatory crossing always preserves the knot, so the cosmetic crossingconjecture can be rephrased that when a crossing change at a crossing c preservesthe knot, then c is nugatory.In [BFKP] Balm-Friedl-Kalfagianni-Powell proved the following constraints forgenus one knots to admit a cosmetic crossing. Theorem 1.1. [BFKP, Theorem 1.1, Theorem 5.1]
Let K be a genus one knot thatadmits a cosmetic crossing. Then K has the following properties. • K is algebraically slice. • For the double branched covering Σ ( K ) of K , H (Σ ( K ); Z ) is finite cyclic. • If K has a unique genus one Seifert surface, ∆ K ( t ) = 1 . In this paper, by using the 2-loop part of the Kontsevich invariant, we provethe cosmetic crossing conjecture for genus one knot with non-trivial Alexanderpolynomial.
Theorem 1.2.
Let K be a genus one knot. If ∆ K ( t ) = 1 , then K satisfies thecosmetic crossing conjecture. For genus one knot K with ∆ K ( t ) = 1 we get an additional constraint for K to admit a cosmetic crossing. Let λ be the Casson invariant of integral homologyspheres and let w ( K ) = V ′′′ K (1)+ V ′′ K (1) be the primitive integer-valued degree3 finite type invariant of K . Here V K ( t ) is the Jones polynomial of K . Theorem 1.3.
Let K be a genus one knot with ∆ K ( t ) = 1 . If λ (Σ ( K )) − w ( K ) , then K satisfies the cosmetic crossing conjecture. The cosmetic crossing conjecture has been confirmed for several cases; 2-bridgeknots [Tor], fibered knots [Kal], knots whose double branched coverings are L-spaceswith square-free 1st homology [LiMo], and some satellite knots [BaKa]. Except thelast satellite cases and the unknot, all the knots mentioned so far, including knotstreated in Theorem 1.1, has non-trivial Alexander polynomial.Theorem 1.3 gives examples of non-satellite knots with trivial Alexander polyno-mial satisfying the cosmetic crossing conjecture. Let K = P ( p, q, r ) be the pretzelknot for odd p, q, r . Obviously, as long as K is non-trivial, g ( K ) = 1. The Alexanderpolynomial of K is∆ K ( t ) = pq + qr + rp + 14 t + − pq − qr − rp + 12 + pq + qr + rp + 14 t − . Hence, for example, the pretzel knot P (4 k + 1 , k + 3 , − (2 k + 1)) has the trivialAlexander polynomial. Corollary 1.4. If k ≡ , , the pretzel knot P (4 k + 1 , k + 3 , − (2 k + 1)) satisfies the cosmetic crossing conjecture. Cosmetic crossing of genus one knot and Seifert surface
We review an argument of [BFKP, Section 2, Section 3] that relates a cosmeticcrossing change and Seifert matrix.A crossing disk D of an oriented knot K is an embedded disk having exactlyone positive and one negative crossing with K . A crossing change can be seen as ε = ± ∂D for an appropriate crossing disk D , and the crossingis nugatory if and only if ∂D bounds an embedded disk in S \ K .Assume that K admits a cosmetic crossing with the crossing disk D . Then as isdiscussed in [BFKP, Section 2], there is a minimum genus Seifert surface S of K such that α := D ∩ S is a properly embedded, essential arc in S .If g ( S ) = 1, such an arc α is non-separating. We take simple closed curves a x , a y of S so that • a x intersects α exactly once. • a x and a y form a symplectic basis of H ( S ; Z ).Then we view K = ∂S as a neighborhood of a x ∪ a y and express K by a framed2-tangle T as depicted in Figure 1. Dα a x a y xy T = Figure 1.
A spine tangle T adapted to the cosmetic crossing OSMETIC CROSSING CONJECTURE FOR GENUS ONE KNOT 3
We call the framed tangle T a spine tangle of K adapted to the cosmetic crossingof a genus one knot K . Let M = (cid:18) n ℓℓ m (cid:19) be the linking matrix of T , where n (resp. m ) is the framing of the strand x (resp. y ) and ℓ is the linking number oftwo strands of T .Let K ′ be a knot obtained from K by crossing change along the crossing disk D .Then S gives rise to a Seifert surface S ′ of K ′ ([BFKP, Proposition 2.1]). K ′ hasa spine tangle presentation T ′ , so that T ′ and T are the same as unframed tangles,and that the linking matrix of T ′ is M ′ = (cid:18) n ± ℓℓ m (cid:19) With respect to the basis { a x , a y } , the Seifert matrix V of K and the Seifertmatrix V ′ of K ′ are given by V = (cid:18) n ℓℓ ± m (cid:19) , V ′ = (cid:18) n ± ℓℓ ± m (cid:19) respectively. Since K and K ′ are the same knot,∆ K ( t ) . = det( V − tV T ) . = det( V ′ − tV ′ T ) . = ∆ K ′ ( t ) . By direct computation, this implies that(2.1) m = 0 . In particular, K is algebraically slice.3. Here we quickly review the 2-loop polynomial. For details, see [Oht]. Let B bethe space of open Jacobi diagram. For a knot K in S , let Z σ ( K ) ∈ B be theKontsevich invariant of K , viewed so that it takes value in B by composing theinverse of the Poincar´e-Birkoff-Witt isomorphism σ : A ( S ) → B .A Jacobi diagram whose edge is labeled by a power series f ( ~ ) = c + c ~ + c ~ + c ~ + · · · represents the Jacobi diagram f ( ~ ) = c + c + c + c + · · · It is known that (the logarithm of) the Kontsevich invariant Z σ ( K ) is written inthe following form [GaKr, Kri].log ⊔ Z σ ( K ) = ✓✒ ✏✑ log (cid:16) sinh( ~ / ~ / (cid:17) − log(∆ K ( e ~ )) + X i :finite ✬✫ ✩✪ p i, ( e ~ ) / ∆ K ( e ~ ) p i, ( e ~ ) / ∆ K ( e ~ ) p i, ( e ~ ) / ∆ K ( e ~ ) + (( ℓ > . Here • ∆ K ( t ) is the Alexander polynomial of K , normalized so that ∆ K (1) = 1and ∆ K ( t ) = ∆ K ( t − ) hold. • log ⊔ is the logarithm with respect to the disjoint union product ⊔ of B ,given bylog ⊔ (1 + D ) = D − D ⊔ D + 13 D ⊔ D ⊔ D + · · · . TETSUYA ITO • p i,j ( e ~ ) is a polynomial of e ~ .Let Θ( t , t , t ; K ) = X ε ∈{± } X σ ∈ S p i, ( t εσ (1) ) p i, ( t εσ (2) ) p i, ( t εσ (3) ) . Here S is the symmetric group of degree 3. The Θ K ( t , t ) ∈ Q [ t ± , t ± ] of a knot K is defined byΘ K ( t , t ) = Θ( t , t , t ; K ) | t = t − t − . The reduced 2-loop polynomial is a reduction of the 2-loop polynomial definedby b Θ K ( t ) = Θ K ( t, t − t − ) ∈ Q [ t ± ] . In general, although Ohtsuki developed fundamental techniques and machineriesthat enable us to compute Θ K ( t , t ), the computation of the 2-loop polynomial ismuch more complicated than the computation of the 1-loop part (i.e., the Alexanderpolynomial). Fortunately, when the knot has genus one, Ohtsuki proved a directformula of Θ K ( t , t ) [Oht, Theorem 3.1]. Consequently he gave the followingformula of the reduced 2-loop polynomial of genus one knots. Theorem 3.1. [Oht, Corollary 3.5]
Let K be a genus one knot expressed by usinga framed 2-tangle T as in Figure 1, and let M = (cid:18) n ℓℓ m (cid:19) be the linking matrix of T . Then b Θ K ( t ) = (cid:16) ( n + m )( d − nm − ℓ ( ℓ + 12 )( ℓ + 1) + 12 v (cid:17)(cid:16) − − d + 13 ( t + t − − (cid:17) − (cid:18) mv xx + nv yy − ( ℓ + 12 ) v xy + 3 v (cid:19) ∆ K ( t ) Here • d = nm − ℓ − ℓ . In particular, ∆ K ( t ) = dt + (1 − d ) + dt − . • v xx , v yy , v xy (resp. v ) are some integer-valued finite type invariant of T whose degree is (resp. ), which do not depend on the framing. Constraint for cosmetic crossings
We prove the Theorem 1.2 and Theorem 1.3 at the same time.
Theorem 4.1.
Let K be a genus one knot. If K admits a cosmetic crossing, then ∆ K ( t ) = 1 and λ (Σ ( K )) − w ( K ) ≡ .Proof. Assume that K is a genus one knot admitting a cosmetic crossing. Weexpress K using a spine tangle T adapted to the cosmetic crossing. Then as wehave seen (2.1), the linking matrix of T is M = (cid:18) n ℓℓ (cid:19) . Moreover, for the knot K ′ obtained by the crossing change, K ′ has a spine tangle T ′ which is identical with T as an unframed tangle with linking matrix is M ′ = (cid:18) n ± ℓℓ (cid:19) . OSMETIC CROSSING CONJECTURE FOR GENUS ONE KNOT 5
Since the finite type invariants v xx , v yy , v xy and v do not depend on the framing,by Theorem 3.1,0 = b Θ K ( t ) − b Θ K ′ ( t )= d ( − − d + 13 ( t + t − − − v yy ( dt + (1 − d ) + dt − )= d (cid:18) − d + 13 − v yy (cid:19) t + d (4 d − v yy (2 d −
1) + d (cid:18) − d + 13 − v yy (cid:19) t − . Therefore(4.1) d (cid:18) − d + 13 − v yy (cid:19) = d (4 d − v yy (2 d −
1) = 0 . If d = 0, by (4.1) d = . Since d ∈ Z , this is a contradiction so we conclude d = 0 and ∆ K ( t ) = 1.Then by (4.1), d = 0 implies v yy = 0. Moreover, since d = nm − ℓ − ℓ = − ℓ ( ℓ +1),we get ℓ = 0 , −
1. Thus by Theorem 3.1, the reduced 2-loop polynomial is b Θ K ( t ) = 12 v (cid:18) − −
13 ( t + t − − (cid:19) − (cid:18) − (cid:18) ℓ + 12 (cid:19) v xy − v (cid:19) hence b Θ K (1) = − v + 4 (cid:18) ℓ + 12 (cid:19) v xy , b Θ K ( −
1) = 4 v + 4 (cid:18) ℓ + 12 (cid:19) v xy . On the other hand, by [Oht, Proposition 1.1] b Θ K (1) = 2 w ( K ) , b Θ K ( −
1) = − V ′ K ( − V K ( − . Since ∆ K ( −
1) = V K ( −
1) = 1, by Mullins’ formula of the Casson-Walker invariant λ w of the double branched coverings [Mul], λ w (Σ ( K )) = − V ′ K ( − V K ( −
1) + σ ( K )4we get b Θ K ( −
1) = 12 λ w (Σ ( K )) . For an integral homology sphere, the Casson invariant λ is twice of the Casson-Walker invariant λ w hence we conclude λ (Σ ( K )) − w ( K ) = b Θ K ( − − b Θ K (1) = 16 v . (cid:3) Proof of Corollary 1.4.
The reduced 2-loop polynomial of genus pretzel knots P ( p, q, r )was given in [Oht, Example 3.6]. In particular, for K = P (4 k + 1 , k + 3 , − (2 k + 1)), b Θ K (1) and b Θ K ( −
1) are given by b Θ K (1) = −
18 (4 k + 2)(4 k + 4)( − k ) , b Θ K ( −
1) = −
124 (4 k + 2)(4 k + 4)( − k )hence λ (Σ ( K )) − w ( K ) = b Θ K ( − − b Θ K (1) = 112 (4 k + 2)(4 k + 4)( − k )= −
16 (2 k + 1)( k + 1) k . TETSUYA ITO
When k ≡ , (2 k +1)( k +1) k Z hence K does not admit cosmetic crossingby Theorem 1.3. (cid:3) Acknowledgement
The author has been partially supported by JSPS KAKENHI Grant Number19K03490,16H02145.
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