L11a196

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Acknowledgement

DrawMorseLink

PD Presentation: X₈₁₉₂ X_10,4,11,3 X_22,10,7,9 X₂₇₃₈ X_20,13,21,14 X_6,12,1,11 X_18,15,19,16 X_4,18,5,17 X_16,6,17,5 X_14,19,15,20 X_12,21,13,22

Gauss Code: {{1, -4, 2, -8, 9, -6}, {4, -1, 3, -2, 6, -11, 5, -10, 7, -9, 8, -7, 10, -5, 11, -3}}

Jones Polynomial: - q^-11/2 + 3q^-9/2 - 5q^-7/2 + 8q^-5/2 - 11q^-3/2 + 12q^-1/2 - 13q^1/2 + 11q^3/2 - 9q^5/2 + 5q^7/2 - 3q^9/2 + q^11/2

A2 (sl(3)) Invariant: q^-18 - 2q^-14 + q^-12 - q^-10 - q^-8 + 3q^-6 - q^-4 + q^-2 + 1 + 3q⁴ - q⁶ + 3q⁸ + 2q¹⁰ - q¹² + 2q¹⁴ - q¹⁸

HOMFLY-PT Polynomial: a^-5z - a^-3z^-1 - 2a^-3z - 2a^-3z³ + a^-1z^-1 + 2a^-1z + a^-1z³ + a^-1z⁵ + az + az³ + az⁵ - 2a³z - 2a³z³ + a⁵z

Kauffman Polynomial: a^-6z² - a^-6z⁴ - 2a^-5z + 4a^-5z³ - 3a^-5z⁵ + 3a^-4z⁴ - 4a^-4z⁶ + a^-3z^-1 - 5a^-3z + 7a^-3z³ - 4a^-3z⁷ - a^-2 + a^-2z⁴ + 3a^-2z⁶ - 4a^-2z⁸ + a^-1z^-1 - 4a^-1z + 5a^-1z³ - a^-1z⁵ + 3a^-1z⁷ - 3a^-1z⁹ + 6z² - 19z⁴ + 20z⁶ - 5z⁸ - z¹⁰ - az + 4az³ - 14az⁵ + 18az⁷ - 6az⁹ + 11a²z² - 32a²z⁴ + 26a²z⁶ - 4a²z⁸ - a²z¹⁰ + a³z - 2a³z³ - 6a³z⁵ + 10a³z⁷ - 3a³z⁹ + 6a⁴z² - 16a⁴z⁴ + 13a⁴z⁶ - 3a⁴z⁸ + a⁵z - 4a⁵z³ + 4a⁵z⁵ - a⁵z⁷

Khovanov Homology:

t^rq^j r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5

j = 12 1

j = 10 2

j = 8 3 1

j = 6 6 2

j = 4 6 4

j = 2 7 5

j = 0 6 7

j = -2 5 6

j = -4 3 6

j = -6 2 5

j = -8 1 3

j = -10 2

j = -12 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=

<< KnotTheory`

Loading KnotTheory` (version of August 30, 2005, 10:15:35)...

In[2]:=
Length[Skeleton[L]]

Out[2]=
2

In[3]:=
Show[DrawMorseLink[Link[11, Alternating, 196]]]

Out[3]=
-Graphics-

In[4]:=
PD[L = Link[11, Alternating, 196]]

Out[4]=
PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[22, 10, 7, 9], X[2, 7, 3, 8], > X[20, 13, 21, 14], X[6, 12, 1, 11], X[18, 15, 19, 16], X[4, 18, 5, 17], > X[16, 6, 17, 5], X[14, 19, 15, 20], X[12, 21, 13, 22]]

In[5]:=
GaussCode[L]

Out[5]=
GaussCode[{1, -4, 2, -8, 9, -6}, > {4, -1, 3, -2, 6, -11, 5, -10, 7, -9, 8, -7, 10, -5, 11, -3}]

In[6]:=
Jones[L][q]

Out[6]=
-(11/2) 3 5 8 11 12 3/2 -q + ---- - ---- + ---- - ---- + ------- - 13 Sqrt[q] + 11 q - 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 11/2 > 9 q + 5 q - 3 q + q

In[7]:=
A2Invariant[L][q]

Out[7]=
-18 2 -12 -10 -8 3 -4 -2 4 6 8 1 + q - --- + q - q - q + -- - q + q + 3 q - q + 3 q + 14 6 q q 10 12 14 18 > 2 q - q + 2 q - q

In[8]:=
HOMFLYPT[Link[11, Alternating, 196]][a, z]

Out[8]=
3 3 1 1 z 2 z 2 z 3 5 2 z z 3 -(----) + --- + -- - --- + --- + a z - 2 a z + a z - ---- + -- + a z - 3 a z 5 3 a 3 a a z a a a 5 3 3 z 5 > 2 a z + -- + a z a

In[9]:=
Kauffman[Link[11, Alternating, 196]][a, z]

Out[9]=
2 -2 1 1 2 z 5 z 4 z 3 5 2 z -a + ---- + --- - --- - --- - --- - a z + a z + a z + 6 z + -- + 3 a z 5 3 a 6 a z a a a 3 3 3 2 2 4 2 4 z 7 z 5 z 3 3 3 5 3 > 11 a z + 6 a z + ---- + ---- + ---- + 4 a z - 2 a z - 4 a z - 5 3 a a a 4 4 4 5 5 4 z 3 z z 2 4 4 4 3 z z 5 > 19 z - -- + ---- + -- - 32 a z - 16 a z - ---- - -- - 14 a z - 6 4 2 5 a a a a a 6 6 7 3 5 5 5 6 4 z 3 z 2 6 4 6 4 z > 6 a z + 4 a z + 20 z - ---- + ---- + 26 a z + 13 a z - ---- + 4 2 3 a a a 7 8 3 z 7 3 7 5 7 8 4 z 2 8 4 8 > ---- + 18 a z + 10 a z - a z - 5 z - ---- - 4 a z - 3 a z - a 2 a 9 3 z 9 3 9 10 2 10 > ---- - 6 a z - 3 a z - z - a z a

In[10]:=
Kh[L][q, t]

Out[10]=
2 1 2 1 3 2 5 3 6 7 + 7 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 12 6 10 5 8 5 8 4 6 4 6 3 4 3 4 2 q t q t q t q t q t q t q t q t 5 6 6 2 4 4 2 6 2 6 3 > ----- + - + ---- + 5 q t + 6 q t + 4 q t + 6 q t + 2 q t + 2 2 t 2 q t q t 8 3 8 4 10 4 12 5 > 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a196
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L11a197

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 196]]]

Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 196]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[22, 10, 7, 9], X[2, 7, 3, 8], > X[20, 13, 21, 14], X[6, 12, 1, 11], X[18, 15, 19, 16], X[4, 18, 5, 17], > X[16, 6, 17, 5], X[14, 19, 15, 20], X[12, 21, 13, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, -8, 9, -6}, > {4, -1, 3, -2, 6, -11, 5, -10, 7, -9, 8, -7, 10, -5, 11, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 3 5 8 11 12 3/2 -q + ---- - ---- + ---- - ---- + ------- - 13 Sqrt[q] + 11 q - 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 11/2 > 9 q + 5 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 2 -12 -10 -8 3 -4 -2 4 6 8 1 + q - --- + q - q - q + -- - q + q + 3 q - q + 3 q + 14 6 q q 10 12 14 18 > 2 q - q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 196]][a, z]
Out[8]=	3 3 1 1 z 2 z 2 z 3 5 2 z z 3 -(----) + --- + -- - --- + --- + a z - 2 a z + a z - ---- + -- + a z - 3 a z 5 3 a 3 a a z a a a 5 3 3 z 5 > 2 a z + -- + a z a
In[9]:=	Kauffman[Link[11, Alternating, 196]][a, z]
Out[9]=	2 -2 1 1 2 z 5 z 4 z 3 5 2 z -a + ---- + --- - --- - --- - --- - a z + a z + a z + 6 z + -- + 3 a z 5 3 a 6 a z a a a 3 3 3 2 2 4 2 4 z 7 z 5 z 3 3 3 5 3 > 11 a z + 6 a z + ---- + ---- + ---- + 4 a z - 2 a z - 4 a z - 5 3 a a a 4 4 4 5 5 4 z 3 z z 2 4 4 4 3 z z 5 > 19 z - -- + ---- + -- - 32 a z - 16 a z - ---- - -- - 14 a z - 6 4 2 5 a a a a a 6 6 7 3 5 5 5 6 4 z 3 z 2 6 4 6 4 z > 6 a z + 4 a z + 20 z - ---- + ---- + 26 a z + 13 a z - ---- + 4 2 3 a a a 7 8 3 z 7 3 7 5 7 8 4 z 2 8 4 8 > ---- + 18 a z + 10 a z - a z - 5 z - ---- - 4 a z - 3 a z - a 2 a 9 3 z 9 3 9 10 2 10 > ---- - 6 a z - 3 a z - z - a z a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 3 2 5 3 6 7 + 7 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 12 6 10 5 8 5 8 4 6 4 6 3 4 3 4 2 q t q t q t q t q t q t q t q t 5 6 6 2 4 4 2 6 2 6 3 > ----- + - + ---- + 5 q t + 6 q t + 4 q t + 6 q t + 2 q t + 2 2 t 2 q t q t 8 3 8 4 10 4 12 5 > 3 q t + q t + 2 q t + q t