Twisted Alexander polynomials with the adjoint action for some classes of knots
aa r X i v : . [ m a t h . G T ] S e p TWISTED ALEXANDER POLYNOMIALS WITH THE ADJOINTACTION FOR SOME CLASSES OF KNOTS
ANH T. TRAN
Abstract.
We calculate the twisted Alexander polynomial with the adjoint action fortorus knots and twist knots. As consequences of these calculations, we obtain the formulafor the nonabelian Reidemeister torsion of torus knots in [Du] and a formula for thenonabelian Reidemeister torsion of twist knots that is better than the one in [DHY]. Introduction
The Alexander polynomial, the first polynomial knot invariant, was discovered byAlexander in 1928 [Al]. It was later interpreted in terms of Reidemeister torsions byMilnor [Mi] and Turaev [Tu]. The twisted Alexander polynomial, a generalization of theAlexander polynomial, was introduced by Lin [Li] for knots in S and by Wada [Wa] forfinitely presented groups. It was also interpreted in terms of Reidemeister torsions byKitano [Ki] and Kirk-Livingston [KL]. As a consequence of this interpretation, one cancalculate certain kinds of Reidemeister torsions of a knot from a finite presentation of itsknot group by applying Fox differential calculus.In this paper we consider the twisted Alexander polynomial with the adjoint action. Theadjoint action, Ad, is the conjugation on the Lie algebra sl ( C ) by the Lie group SL ( C ).Suppose K is a knot and G K its knot group. For each representation ρ : G K → SL ( C ),the composition Ad ◦ ρ : G K → SL ( C ) is a representation and hence, following [Wa],one can define a rational function ∆ Ad ◦ ρK ( t ), called the twisted Alexander polynomial withthe adjoint action associated to ρ . The twisted Alexander polynomial ∆ Ad ◦ ρK ( t ) has beencalculated for just a few knots [DY]. The purpose of this paper is to calculate ∆ Ad ◦ ρK ( t )for torus knots and twist knots, see Theorems 2.3 and 2.5.The paper is organized as follows. In Section 2 we review some backgrounds on thetwisted Alexander polynomial with the adjoint action and state the main results, The-orems 2.3 and 2.5, about the formulas for the twisted Alexander polynomial with theadjoint action for torus knots and twist knots. We give proofs of Theorems 2.3 and 2.5in Sections 3 and 4 respectively.2. The twisted Alexander polynomial with the adjoint action
Twisted Alexander polynomials.
Let K be a knot and G K = π ( S \ K ) its knotgroup. We fix a presentation G K = h a , . . . , a ℓ | r , . . . , r ℓ − i . (This might not be a Wirtinger representation, but must be of deficiency one.) Mathematics Subject Classification . 57M27.
Key words and phrases. twisted Alexander polynomial, Reidemeister torsion, adjoint action, torusknot, twist knot.
Let f : G K → H ( S \ K, Z ) ∼ = Z = h t i be the abelianization homomorphism and ρ : G K → SL k ( C ) a representation. These maps naturally induce two ring homomorphisms e f : Z [ G K ] → Z [ t ± ] and e ρ : Z [ G K ] → M ( k, C ), where Z [ G K ] is the group ring of G K and M ( k, C ) is the matrix algebra of degree k over C . Then e ρ ⊗ e f : Z [ G K ] → M ( k, C [ t ± ])is a ring homomorphism. Let F ℓ be the free group on generators a , . . . , a ℓ and Φ : Z [ F ℓ ] → M ( k, C [ t ± ]) the composition of the surjective map Z [ F ℓ ] → Z [ G K ] induced bythe presentation of G K and the map e ρ ⊗ e f : Z [ G K ] → M ( k, C [ t ± ]).We consider the ( ℓ − × ℓ matrix M whose ( i, j )-component is the k × k matrixΦ (cid:18) ∂r i ∂a j (cid:19) ∈ M (cid:0) k, C [ t ± ] (cid:1) , where ∂∂a denotes the Fox derivative. For 1 ≤ j ≤ ℓ , let M j be the ( ℓ − × ( ℓ −
1) matrixobtained from M by removing the j th column. We regard M j as a k ( ℓ − × k ( ℓ − C [ t ± ]. Then Wada’s twisted Alexander polynomial of the knot K associated to the representation ρ : G K → SL k ( C ) is defined to be the rational function∆ ρK ( t ) = det M j det Φ(1 − a j ) . It is defined up to a factor t km ( m ∈ Z ), see [Wa].2.2. The twisted Alexander polynomial with the adjoint action.
The adjointaction, Ad, is the conjugation on the Lie algebra sl ( C ) by the Lie group SL ( C ). For A ∈ SL ( C ) and g ∈ sl ( C ) we have Ad A ( g ) = AgA − . For each representation ρ : G K → SL ( C ), the composition Ad ◦ ρ : G K → SL ( C ) is a representation and henceone can define the twisted Alexander polynomial ∆ Ad ◦ ρK ( t ). We call ∆ Ad ◦ ρK ( t ) the twistedAlexander polynomial with the adjoint action associated to ρ .In this paper we are interested in the twisted Alexander polynomial with the adjointaction associated to irreducible/non-abelian SL ( C )-representations. Remark 2.1.
It is known that ∆ Ad ◦ ρK ( t ) coincides with the nonabelian Reidemeister tor-sion polynomial T ρK ( t ) [Ki, KL]. As a consequence of this identification, one can calculatethe nonabelian Reidemeister torsion T ρK for any longitude-regular SL ( C )-representation ρ from a finite presentation of the knot group of K , by applying Fox differential calculusand the following formula(2.1) T ρK = − lim t → T ρK ( t ) t − T ρK ( t ) and T ρK .2.3. Torus knots.
Let K be the ( p, q )-torus knot.The standard presentation for the knotgroup of K is G K = h c, d | c p = d q i . Choose a pair ( r, s ) of natural numbers such that ps − qr = 1. Then µ = c − r d s is a meridian of K . Note that the abelian homomorphism f : G K → H ( S \ K ; Z ) ∼ = Z = h t i sends c and d to t q and t p respectively.A representation ρ : G K → SL ( C ) is called irreducible if there is no proper invariantline in C under the action of ρ ( G K ). Let R irr ( G K ) be the set of irreducible SL ( C )-representations of G K and ˆ R irr the set of conjugacy classes of representations in R irr ( G K ).According to [Jo, Prop. 34], we have the following description of ˆ R irr ( G K ). See also[Kl, Le] for similar results. WISTED ALEXANDER POLYNOMIALS WITH THE ADJOINT ACTION 3
Proposition 2.2. ˆ R irr ( G K ) consists of ( p − q − / components, which are determinedby the following data, denoted by ˆ R irr k,l ( G K ) : (1) 0 < k < p , < l < q , and k ≡ l (mod 2) . (2) For every [ ρ ] ∈ ˆ R irr k,l ( G K ) , we have ρ ( c p ) = ρ ( d q ) = ( − k I . Moreover, tr ρ ( c ) =2 cos (cid:0) πkp (cid:1) , tr ρ ( d ) = 2 cos (cid:0) πlq (cid:1) , and tr ρ ( µ ) = 2 cos π (cid:0) rkp ± slq (cid:1) .In particular, ˆ R irr k,l ( G K ) is parametrized by tr ρ ( µ ) and has complex dimension one. Then we have the following.
Theorem 2.3.
Let K be the ( p, q ) -torus knot. Suppose ρ : G K → SL ( C ) is a represen-tation such that [ ρ ] ∈ ˆ R irr k,l . Then ∆ Ad ◦ ρK ( t ) = ( t pq − ( t p − t q − t q − πkp ) t q + 1)( t p − πlq ) t p + 1) . It is known that for a torus knot every irreducible SL ( C )-representation is longitude-regular, and hence one can define the non-abelian Reidemeister torsion T ρK , see [Po, Du].Theorem 2.3 and equation (2.1) imply the following. Corollary 2.4 ([Du]) . Let K be the ( p, q ) -torus knot. Suppose ρ : G K → SL ( C ) is arepresentation such that [ ρ ] ∈ ˆ R irr k,l . Then T ρK = − p q
16 sin (cid:0) πkp (cid:1) sin (cid:0) πlq (cid:1) . Twist knots.
Let J ( k, l ) be the link in Figure 1, where k, l denote the numbers ofhalf twists in the boxes. Positive (resp. negative) numbers correspond to right-handed(resp. left-handed) twists. Note that J ( k, l ) is a knot if and only if kl is even, and is thetrivial knot if kl = 0. Furthermore, J ( k, l ) ∼ = J ( l, k ) and J ( − k, − l ) is the mirror image of J ( k, l ). Hence we only consider J ( k, n ) for k > | n | >
0. When k = 2, J (2 , n ) isthe twist knot. For more information about J ( k, l ), see [HS]. k l Figure 1.
The link J ( k, l ).Consider K = J (2 , n ). The knot group of K is G K = h a, b | w n a = bw n i where a, b are meridians and w = ba − b − a , see [HS]. A representation ρ : G K → SL ( C ) is callednonabelian if ρ ( G K ) is a nonabelian subgroup of SL ( C ).Let x = tr ρ ( a ) = tr ρ ( b ) and y = tr ρ ( ab − ). Then we have the following. ANH T. TRAN
Theorem 2.5.
Let K be the twist knot J (2 , n ) . Suppose ρ : G K → SL ( C ) is a non-abelian representation. Then ∆ Ad ◦ ρK ( t ) = t − y + 2 − x )( y − yx + x ) × (cid:18) nt + (2 n − y + yx − nx ( x − y − yx + 2 x t + n (cid:19) . Theorem 2.5 and equation (2.1) imply the following.
Corollary 2.6.
Let K be the twist knot J (2 , n ) . Suppose ρ : G K → SL ( C ) is alongitude-regular representation. Then T ρK = − y + 2 − x )( y − yx + x ) (cid:18) (2 n − y + yx − nx ( x − y − yx + 2 x + 2 n (cid:19) . Remark 2.7.
The nonabelian Reidemeister torsion for twist knots was calculated in[DHY]. However, the formula in Corollary 2.6 is better.3.
Proof of Theorem 2.3
Recall that K is the ( p, q )-torus knot and G K = h c, d | c p = d q i the standard presenta-tion of its knot group. Suppose ρ : G K → SL ( C ) is a representation such that [ ρ ] ∈ ˆ R irr k,l .Proposition 2.2 implies that the matrices ρ ( c ) and ρ ( d ) are respectively conjugate to " e i πkp e − i πkp and " e i πlq e − i πlq . By conjugation if necessary, we may assume that ρ ( c ) = (cid:20) α α − (cid:21) and ρ ( d ) is conjugateto (cid:20) β β − (cid:21) where α = e i πkp and β = e i πlq .Let { E, H, F } be the following usual C -basis of sl ( C ): E = (cid:20) (cid:21) , H = (cid:20) − (cid:21) , F = (cid:20) (cid:21) . Then the adjoint actions of c and d in the basis { E, H, F } of sl ( C ) are respectivelygiven by the matrices C = Ad ρ ( c ) and D = Ad ρ ( d ) , where C = diag( α , , α − ) and D isconjugate to diag( β , , β − ).We have ∂∂c c p d − q = 1 + c + · · · + c p − , and hence∆ Ad ◦ ρK ( t ) = det Φ( ∂∂c c p d − q )det Φ( d − I + t q C + t q C + · · · + t ( p − q C p − )det( t p D − I )= (1 + α t q + · · · + α p − t ( p − q )(1 + α − t q + · · · + α − p − t ( p − q ) t p − ( β + β − ) t p + 1 × t q + · · · + t ( p − q t p − . WISTED ALEXANDER POLYNOMIALS WITH THE ADJOINT ACTION 5
Since α p = 1, we have1 + α t q + · · · + α p − t ( p − q = ( α t q ) p − α t q − t pq − α t q − . Similarly, 1 + α − t q + · · · + α − p − t ( p − q = t pq − α − t q − . Hence∆ Ad ◦ ρK ( t ) = ( t pq − ( t p − t q − t q − ( α + α − ) t q + 1)( t p − ( β + β − ) t p + 1) . Theorem 2.3 follows, since α + α − = 2 cos( πkp ) and β + β − = 2 cos( πlq ). Remark 3.1.
The above proof is similar to that of [KM, Thm. 1.1].4.
Proof of Theorem 2.5
Recall that K is the twist knot J (2 , n ) and G K = h a, b | w n a = bw n i its knot group,where a, b are meridians and w = ba − b − a .4.1. Nonabelian representations.
Suppose ρ : G K → SL ( C ) is a nonabelian repre-sentation. Taking conjugation if necessary, we can assume that ρ has the form ρ ( a ) = (cid:20) √ s √ s √ s (cid:21) and ρ ( b ) = (cid:20) √ s −√ s u √ s (cid:21) where ( s, u ) ∈ C ∗ × C satisfies the Riley equation φ K ( s, u ) = 0, see [Ri]. Note that x = tr ρ ( a ) = √ s + √ s and y = tr ρ ( ab − ) = u + 2.Let γ = tr ρ ( w ) = 2 + 2 u − us − su + u . By [DHY] we have φ K ( s, u ) = ( s + s − − − u ) ξ n + − ξ n − ξ + − ξ − − ξ n − − ξ n − − ξ + − ξ − where ξ ± are eigenvalues of ρ ( w ), i.e. ξ + ξ − = 1 and ξ + + ξ − = γ .Let X = ξ n + − ξ n − ξ + − ξ − , Y = ξ n − − ξ n − − ξ + − ξ − . Since φ K ( s, u ) = 0, we have Y = ( s + s − − − u ) X . Moreover, it is easy to see that X − γXY + Y = 1. Hence we have the following. Lemma 4.1. X = 11 − ( s + s − − − u ) γ + ( s + s − − − u ) . ANH T. TRAN
Adjoint action matrices.
Recall that { E, H, F } is the following usual C -basis of sl ( C ): E = (cid:20) (cid:21) , H = (cid:20) − (cid:21) , F = (cid:20) (cid:21) . The adjoint actions of a and b in the basis { E, H, F } of sl ( C ) are given by the followingmatrices: A = Ad ρ ( a ) = s − − s − s − s − , B = Ad ρ ( b ) = s su − su − u s − . Let W = Ad ρ ( w ) . Note that the SL ( C )-matrix ρ ( w ) can be diagonalized by Q = (cid:20) u + 1 − s − u + 1 − s − − su − ξ + − su − ξ − (cid:21) . Explicitly, Q − ρ ( w ) Q is the diagonal matrix diag( ξ + , ξ − ).Let α = 1 − su − ξ + , β = 1 − su − ξ − and δ = u + 1 − s − . With respect to the basis { E, H, F } of sl ( C ), the matrix of the adjoint action of Q is P = Ad Q = 1 α − β − δ δ δα − ( α + β ) − βα /δ − αβ/δ − β /δ . Then P − W P is the diagonal matrix diag( ξ , , ξ − ).Let Ω = I + W − + · · · + W − ( n − . We have the following. Proposition 4.2.
Ω = 1 s u (1 − s + s − su )( − s + u − su + s u − su ) ω ω ω ω ω ω ω ω ω WISTED ALEXANDER POLYNOMIALS WITH THE ADJOINT ACTION 7 where ω = s u (cid:8) (1 − s + s − su )(2 − s + 2 s + u − su + s u − s u − su + 3 s u − s u + s u − s u ) X − ns ( − s + su ) (cid:9) ,ω = − su ( − s + su ) (cid:8) (1 − s + s − su ) × ( − − s + 2 su − s u + s u − s u ) X − ns ( − s + s + su ) (cid:9) ,ω = − ( − s + su ) (cid:8) (1 − s + s − su ) × ( − s + u − su + s u − su ) X − ns (cid:9) ,ω = s u ( − s + su ) (cid:8) (1 − s + s − su ) × ( − − s + 2 su − s u + s u − s u ) X − ns ( − s + s + su ) (cid:9) ,ω = − su (cid:8) − s + s − su )( − s + su ) ( − s + u − su + s u − su ) X − nsu ( − s + s + su ) (cid:9) ,ω = − u ( − s + su ) (cid:8) (1 − s + s − su )( − s + su )( − s + s + u − su + s u − su ) X − ns ( − s + s + su ) (cid:9) ,ω = − s u ( − s + su ) (cid:8) (1 − s + s − su ) × ( − s + u − su + s u − su ) X − ns (cid:9) ,ω = 2 su ( − s + su ) (cid:8) (1 − s + s − su )( − s + su )( − s + s + u − su + s u − su ) X − ns ( − s + s + su ) (cid:9) ,ω = u ( − s + su ) (cid:8) (1 − s + s − su )( − s + 2 s + 4 su − s u + 3 s u − u + 4 su − s u + 3 s u + 2 su − s u + s u − s u ) X − ns ( − s + su ) (cid:9) . Proof.
Let d = ξ n − − + ξ n − = 2 X − γY,d = α ξ n − − + β ξ n − = (1 − su )(2 X − γY ) − γX + ( γ − Y,d = α ξ n − + β ξ n − − = (1 − su )(2 X − γY ) − γX + 2 Y,d = α ξ n − − + β ξ n − = (1 − su ) (2 X − γY ) − − su )( γX − ( γ − Y )+ ( γ − X − γ ( γ − Y,d = α ξ n − + β ξ n − − = (1 − su ) (2 X − γY ) − − su )( γX − Y )+ ( γ − X − γY. Since P − W P is the diagonal matrix diag( ξ , , ξ − ), we haveΩ = P diag( ξ n − − X, n, ξ n − X ) P − = 1( α − β ) − αβn + d X δ ( − ( α + β ) n + d X ) δ (2 n − d X ) αβδ (( α + β ) n − d X ) ( α + β ) n − αβd X δ ( − ( α + β ) n + d X )( αβδ ) (2 n − d X ) αβδ (( α + β ) n − d X ) − αβn + d X . Proposition 4.2 follows from the above equation and Y = ( s + s − − − u ) X . (cid:3) ANH T. TRAN
Proof of Theorem 2.5.
We focus on the case n >
0. The case n < ∂∂a w n aw − n b − = w n (1 + (1 − a )(1 + w − + · · · + w − ( n − )( a − − a − b ))and hence∆ Ad ◦ ρK ( t ) = det Φ( ∂∂a w n aw − n b − )det Φ( b −
1) = det( I + ( I − tA )Ω( t − A − − A − B ))det( tB − I ) . Then, by Lemma 4.1 and Proposition 4.2, we have∆ Ad ◦ ρK ( t ) = s ( t − − s − u )(1 − s + s − su )( − s + su )( − s + u − su + s u − su ) t × (cid:8) ns ( − s + u − su + s u − su ) t + (2 n − s + 4 ns − ns − s + 4 ns + 2 ns − su + 2 s u − ns u − s u + s u − ns u ) t + ns ( − s + u − su + s u − su ) (cid:9) = t − − s + s − u )( − s + s − u + us + su − u ) t × (cid:8) nt + 1 − − u + us + su − u (cid:0) − n + 2 ns − s + 4 ns − s + 4 ns + 2 ns + 2 u − nu − us − su + u − nu (cid:1) t + n (cid:9) . By substituting s + s = x − s + s = x − x + 2 and u = y − References [Al] J. Alexander,
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