On the volume and the Chern-Simons invariant for the hyperbolic …jyham/KMS2017.pdf ·...

OutlineNeumann’s Methods

The complex volume of alternating knotsThe complex volume of J(2n,−2m) knots

On the volume and the Chern-Simons invariantfor the hyperbolic alternating knot orbifolds.

Dedicated to Professor Joan Birman on the occasion of herninetieth birthday.

J. Ham, H. Kim, J. Lee, and S. Yoon

Apr. 29, 2017

J. Ham, H. Kim, J. Lee, and S. Yoon Volume, Chern-Simons, alternating knot orbifolds, J(2n,−2m)



Neumann’s MethodsAn extended version of Rodger’s dilogarithm functionNeumann’s complex volume formulaZickert’s methods in [7]Neumann’s methods versus Zickert’s methods

The complex volume of alternating knotsPotential functionThe hyperbolicity equations of alternating knot orbifoldsThe complex volume of alternating knot orbifolds

The complex volume of J(2n,−2m) knotsThe fundamental group of J(2n,−2m) knotsRiley-Mednykh polynomial of J(2n,−2m)The shape of J(2n,−2m) knot orbifoldsThe complex volume of J(2,−2) knot orbifolds




An extended version of Rodger’s dilogarithm functionNeumann’s complex volume formulaZickert’s methods in [7]Neumann’s methods versus Zickert’s methods

Some preliminary figures I [6]





Some preliminary figures II [6]

Let K be an alternating knot and M be the r -fold cyclic coveringof S3\K . Let M ′ be the completion of M obtained by a hyperbolic(1, 0) Dehn filling.





Some preliminary figures II [6]





An extended version of Rodger’s dilogarithm function

Let P(C) be the extended pre-Bloch group and B(C) be theextended Bloch group. Define

R(z ; p, q) = R(z) +πi

2(p log (1− z) + q log z)− π2

6

= Li2(z)− π2

6+

1

2qπi log z +

1

2log (1− z)(log z + pπi)

where R is the Rogers dilogarithm function

R(z) =1

2log z log (1− z)−

∫ z

0

log (1− t)

tdt

and Li2 is the dilogarithm function

Li2(z) = −∫ z

0

log (1− t)

tdt.





Theorem 14.7 of [5]

Consider the degree one triangulation of M. Then there existsflattenings [xi ; pi , qi ] of the ideal simplices which satisfy theconditions

(i) parity along normal paths is zero;

(ii) log-parameter about each edge is zero;

(iii) log-parameter along any normal path in the neighborhood of a0-simplex that represents an unfilled cusp is zero;

(iv) log-parameter along a normal path in the neighborhood of a0-simplex that represents a filled cusp is zero if the path isnull-homotopic in the added solid torus.

For any such choice of flattenings we have

β(M ′) =∑i

εi [xi ; pi , qi ] ∈ B(C)





Theorem 14.7 of [5]-continued

so

i(vol + i cs)(M ′) =∑i

εiR(xi ; pi , qi ).





Since the holonomy is boundary parabolic, we can apply Zickert’smethods in [7] to our octahedral triangulation to find flatteningparameters [xi ; pi , qi ] which satisfy the Neumann’s conditionsexcept the condition (i) [7, Theorem 6.5]. Denote M ′ = Mr (K ).Without the condition (i), the Neumann’s complex volumeequation will still give the complex volume at worst moduloπ2/2 [5, Lemma 11.3]. But since with our triangulation theNeumann’s equation gives the complex volume modulo π2 atO(K ,∞) when regarded as a knot complement [1, Theorem 1.2],it will give the complex volume modulo π2 at Mr (K ) for r odd andmodulo 2π2 at Mr (K ) for r even and hence π2/r at O(K , r) for rodd and modulo 2π2/r at O(K , r) for r even [4, Theorem 3.9].With our octahedral triangulation, the Neumann’s complex volumeequation gives the equation of Theorem 3.2





Neumann’s methods versus Zickert’s methods

λ : H3(PSL(2,C);Z)∼= // B(C) .

Neumann showed that any complete hyperbolic 3-manifold M offinite volume has a natural fundamental class in H3(PSL(2,C);Z).

Ψ : H3(PSL(2,C),P)→ B(C). (1)

H3(PSL(2,C),P;Z)s // H3(PSL(2,C);Z)

∼= // B(C)

where s is a natural splitting of the map

H3(PSL(2,C);Z) // H3(PSL(2,C),P;Z) .




Potential functionThe hyperbolicity equations of alternating knot orbifoldsThe complex volume of alternating knot orbifolds

the potential function of a crossing

Let D be a reduced planar projection of an alternating knot K withn/2 crossings with n even. For each crossing of D, we define thepotential function of a crossing as the sum of four terms as in thefollowing figure.

zd zc

za zb

Li2

(zbza

)− Li2

(zbzc

)+ Li2

(zdzc

)− Li2

(zdza

)

Figure: The potential function for a crossing





the potential function of the diagram of D

Definition 3.1.The potential function V (z1, . . . , zn) of the diagram D is definedas the summation of all potential functions of the n/2 crossings.

Note: Li2 is the dilogarithm function

Li2(z) = −∫ z

0

log (1− t)

tdt.





Example - 41 knot

z1 z2

z ′3

z ′4

z3 z4

z5 z6

Figure: A diagram of a figure 8 knot





Example - 41 knot - Potential Function

V (z1, z2, z3, z4, z5, z6, z′3, z′4)

= Li2

(z(1)

z(2)

)− Li2

(z(1)

z(3)

)+ Li2

(z(4)

z(3)

)− Li2

(z(4)

z(2)

)+ Li2

(z(3)

z(4)

)− Li2

(z(3)

z(5)

)+ Li2

(z(6)

z(5)

)− Li2

(z(6)

z(4)

)+ Li2

(z(5)

z ′(3)

)− Li2

(z(5)

z(1)

)+ Li2

(z ′(4)

z(1)

)− Li2

(z ′(4)

z ′(3)

)+ Li2

(z ′(3)

z(6)

)− Li2

(z ′(3)

z ′(4)

)+ Li2

(z(2)

z ′(4)

)− Li2

(z(2)

z(6)

).





The hyperbolicity equations of alternating knot orbifolds

Define

H :=

{exp

(zk∂V

∂zk

)= exp

(±2πi

r

)| k = 1, . . . , n

}where + corresponds to Type I and − corresponds to Type II ofthe following figure.

zk zk

Figure: The side of Type I (left) and the side of Type II (right)





Then H gives the following set of equations which we also write Hfor notational convenience:{

exp

(r∑1

zk∂V

∂zk

)= 1 | k = 1, . . . , n

}which gives the set of hyperbolicity equations of r -fold cycliccovering, M, of S3\K .





Example - 41 knot - Hyperbolicity Equation

H :=

{exp

(zk∂V

∂zk

)= exp

(±2

πi

r

) ∣∣ k = 1, . . . , 8, z7 = z ′3, z8 = z ′4

}.

For example,

exp

(z4∂V

∂z4

)=

(1− z(3)

z(4)

)(1− z(4)

z(2)

)(

1− z(6)z(4)

)(1− z(4)

z(3)

) ,

exp

(z1∂V

∂z1

)=

(1− z(1)

z(3)

)(1− z ′(4)

z(1)

)(

1− z(1)z(2)

)(1− z(5)

z(1)

) .J. Ham, H. Kim, J. Lee, and S. Yoon Volume, Chern-Simons, alternating knot orbifolds, J(2n,−2m)




Example - 41 knot - Cusp Shape Type I

Blue arrows: exp(z4

∂V∂z4

)z5 z1z4

z3

z6

z3

z2z4z3

z2z4

z3z4

z4z6

s





Example - 41 knot - Cusp Shape Type II

Blue arrows: exp(z1

∂V∂z1

)z4 z′3z1

z3

z2

z5

z′4z3z1

z1z2

z1z5

z ′4z1

s





The complex volume of alternating knot orbifolds

Theorem 3.2.[2] Let K be a hyperbolic alternating knot with a reduced diagram

and V (z1, ..., zn) be the potential function of the diagram. Thenfor any z ∈ S which will give the maximal volume, we have

i(vol + i cs)(O(K , r)) = V (z1, ..., zn)−n∑

k=1

(zk∂V

∂zk

)log zk(

modπ2

rfor r odd and mod

2π2

rfor r even

).





Example - 41 knot - Complex Volume of O(41, 4)

z1 = 2, z2 = 1, z3 = −1, z4 ≈ 0.1830127 + 1.3169873i ,

z5 ≈ 0.7616864− 0.1915678i , z6 ≈ 1.5 + 2.1339746i ,

z ′3 ≈ 4.7320508− 2.7320508i , z ′4 = 2.0638559− 0.4127712i .

i(vol + i cs)(O(41, 4)) ≈ 4.9348022 + 0.5074708i

where 4.9348022 ≈ 2π2/4.




The fundamental group of J(2n,−2m) knotsRiley-Mednykh polynomial of J(2n,−2m)The shape of J(2n,−2m) knot orbifoldsThe complex volume of J(2,−2) knot orbifolds

J(2n,−2m) knots (C (2m, 2n) knots).

z1 z2z3 z4

z5 z6...z2n−1 z2nz2n+1 z2n+2

z ′1

z ′2

z ′3

z ′4

z ′2m+1

z ′2m+2

. . .

Figure: A diagram of J(n,−m) with segment variables.





The fundamental group of J(2n,−2m) knots

Proposition 4.1.

LetX 2m2n be S3\J(2n,−2m).

π1(X 2m2n ) =

⟨s, t | swmt−1w−m = 1

⟩,

where w = (t−1s)n(ts−1)n.

I Slope of J(2n,−2m)= 2m/(4nm + 1) ∼ (2n(2m − 1) + 1) /(4nm + 1) [3].




The fundamental group of J(2n,−2m) knotsRiley-Mednykh polynomial of J(2n,−2m)The shape of J(2n,−2m) knot orbifoldsThe complex volume of J(2,−2) knot orbifolds(

PSL(2,C ),H3)

structure of X 2m2n (α)

Given a set of generators, s, t, of the fundamental group for

π1(X 2m2n ) and for M = e

iα2 , let

S =

[M 10 M−1

], T =

[M 0

2−M2 −M−2 − x M−1

],

and c =

[0 −(

√2−M2 −M−2 − x)−1√

2−M2 −M−2 − x 0

].

Then tr(T−1S) = x + M2 + M−2 = tr(TS−1). Letv = x + M2 + M−2, ρ(w) = W , and z = Tr(W ).





Riley-Mednykh polynomial of J(2n,−2m)

Lemma 4.2.ρ is a representation of π1(X 2m

2n ) if and only if x is a root of thefollowing Riley-Mednykh polynomial of J(2n,−2m),

φ2m2n (x ,M) = [−1 + xSn−1(v) (Sn(v) + (1− v)Sn−1(v))]Sm−1(z)

+ Sm(z).

where Sk(ξ) be the Chebychev polynomials defined by S0(ξ) = 1,S1(ξ) = ξ and Sk(ξ) = ξSk−1(ξ)− Sk−2(ξ) for all integers k .





M2-deformed solutions for J(2n,−2m)

Theorem 4.3.An M2-deformed solution (z1, · · · , z2n+2, z

′1, · · · , z ′2m+2) for the

J(n,−m) knot is given by

z2j−1 =Pj−1Qj+1

, z2j =Pj

Qj, z ′2j ′−1 =

P ′j ′−1Q ′j ′+1

, and z ′2j ′ =P ′j ′

Q ′j ′

for 1 ≤ j ≤ n + 1, 1 ≤ j ′ ≤ m + 1 where Λ satisfies

B ′m+1 + B ′mBn−1 = 0.





M2-deformed solutions for J(2n,−2m)-continued I

For a rational polynomial P of M we define P by a rational

polynomial obtained from P by replacing M with1

M. Let (Bj)j∈Z

be the sequence defined by the following recurrence relations{Bj+1 =

√ΛBj + M2Bj−1 for odd j

Bj+1 =√

ΛBj + M−2Bj−1 for even j(2)

with initial conditions B0 = 0, B1 = 1. Let sequences (Pj)j∈Z and(Qj)j∈Z be also defined by the same recurrence relations as Bj with

initial conditions P0 = z1(z2 −z3M2

), P1 =√

Λz2z3 and

Q1 =√

Λz3, Q2 = z2 −z3M2

.





M2-deformed solutions for J(2n,−2m)-continued II

Let (B ′j )j∈Z be the sequence defined by the following recurrencerelations:{

B ′j+1 = (Bn+1 − Bn−1)B ′j + M−2B ′j−1 for odd j

B ′j+1 = (Bn+1 − Bn−1)B ′j + M2B ′j−1 for even j(3)

with initial condition B ′0 = 0, B ′1 = 1. Let sequences (P ′j )j∈Z and(Q ′j )j∈Z be defined by the same recurrence relations as B ′j withinitial conditions P ′0 = Pn, P

′1 = P0 and Q ′1 = Q2, Q

′2 = Qn+2.





Riley-Mednykh polynomial versus the polynomial inTheorem 4.3

Λ =(M2z2 − z3)(z4 −M2z1)

M2z2z3,

which is the same Λ of the polynomial in Theorem 4.3 is the sameas x in the Riley-Mednykh Polynomial.





The complex volume of O(J(2,−2), r)

r Λ i(vol + i cs)(O(J(2n,−2m), r))

4 0.5000000+0.8660254 i 4.93480220+0.50747080 i5 0.1909830+0.9815933 i 3.94784176+0.93720685 i6 1.000000 i 3.28986813+1.22128746 i7 -0.1234898+0.9923458 i 2.81988697+1.41175465 i8 -0.2071068+0.9783183 i 2.46740110+1.54386328 i9 -0.2660444+0.9639608 i 2.19324542+1.63860068 i

10 -0.3090170+0.9510565 i 1.97392088+1.70857095 i





Jinseok Cho, Hyuk Kim, and Seonwha Kim.Optimistic limits of kashaev invariants and complex volumes ofhyperbolic links.J. Knot Theory Ramifications, 23(9), 2014.

Ji-Young Ham, Hyuk Kim, Joongul Lee, and Seokbeom Yoon.On the volume and the Chern-Simons invariant for thehyperbolic alternating knot orbifolds.www.math.snu.ac.kr/~jyham, 2017.

Ji-Young Ham and Joongul Lee.Explicit formulae for Chern-Simons invariants of the hyperbolicJ(2n,−2m) knot orbifolds.www.math.snu.ac.kr/~jyham, 2017.

Hugh M. Hilden, Marıa Teresa Lozano, and Jose MarıaMontesinos-Amilibia.


www.math.snu.ac.kr/~jyham

www.math.snu.ac.kr/~jyham




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Christian K. Zickert.The volume and Chern-Simons invariant of a representation.Duke Math. J., 150(3):489–532, 2009.


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