L11n116

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Acknowledgement

DrawMorseLink

PD Presentation: X₆₁₇₂ X_7,17,8,16 X_20,17,21,18 X_18,13,19,14 X_14,19,15,20 X_4,21,1,22 X_10,5,11,6 X_12,3,13,4 X_22,11,5,12 X_2,9,3,10 X_15,9,16,8

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

Jones Polynomial: q^-21/2 - 4q^-19/2 + 7q^-17/2 - 11q^-15/2 + 14q^-13/2 - 14q^-11/2 + 13q^-9/2 - 11q^-7/2 + 6q^-5/2 - 3q^-3/2

A2 (sl(3)) Invariant: - q^-32 + 2q^-30 + q^-28 - q^-26 + 4q^-24 - 2q^-22 - q^-20 - q^-18 - 3q^-16 + 3q^-14 - q^-12 + 4q^-10 + 3q^-8 - q^-6 + 3q^-4

HOMFLY-PT Polynomial: - 2a³z^-1 - 5a³z - 3a³z³ + 4a⁵z^-1 + 6a⁵z + 5a⁵z³ + 2a⁵z⁵ - 3a⁷z^-1 - 3a⁷z + a⁷z⁵ + a⁹z^-1 - a⁹z³

Kauffman Polynomial: - 2a³z^-1 + 7a³z - 6a³z³ + 2a⁴ - 3a⁴z² - 3a⁴z⁶ - 4a⁵z^-1 + 15a⁵z - 22a⁵z³ + 14a⁵z⁵ - 7a⁵z⁷ + 3a⁶ - 5a⁶z² - a⁶z⁴ + 7a⁶z⁶ - 6a⁶z⁸ - 3a⁷z^-1 + 12a⁷z - 27a⁷z³ + 31a⁷z⁵ - 10a⁷z⁷ - 2a⁷z⁹ + 3a⁸ - 2a⁸z² - 7a⁸z⁴ + 22a⁸z⁶ - 11a⁸z⁸ - a⁹z^-1 + 4a⁹z - 19a⁹z³ + 28a⁹z⁵ - 7a⁹z⁷ - 2a⁹z⁹ + a¹⁰ - a¹⁰z² - 4a¹⁰z⁴ + 11a¹⁰z⁶ - 5a¹⁰z⁸ - 8a¹¹z³ + 11a¹¹z⁵ - 4a¹¹z⁷ - a¹²z² + 2a¹²z⁴ - a¹²z⁶

Khovanov Homology:

t^rq^j r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0

j = -2 3

j = -4 4 1

j = -6 7 2

j = -8 6 4

j = -10 8 7

j = -12 7 7

j = -14 4 7

j = -16 3 7

j = -18 1 4

j = -20 3

j = -22 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, NonAlternating, 116]]]

Out[3]=
-Graphics-

In[4]:=
PD[L = Link[11, NonAlternating, 116]]

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

In[5]:=
GaussCode[L]

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

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

Out[6]=
-(21/2) 4 7 11 14 14 13 11 6 3 q - ----- + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q q

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

Out[7]=
-32 2 -28 -26 4 2 -20 -18 3 3 -12 4 -q + --- + q - q + --- - --- - q - q - --- + --- - q + --- + 30 24 22 16 14 10 q q q q q q 3 -6 3 > -- - q + -- 8 4 q q

In[8]:=
HOMFLYPT[Link[11, NonAlternating, 116]][a, z]

Out[8]=
3 5 7 9 -2 a 4 a 3 a a 3 5 7 3 3 5 3 ----- + ---- - ---- + -- - 5 a z + 6 a z - 3 a z - 3 a z + 5 a z - z z z z 9 3 5 5 7 5 > a z + 2 a z + a z

In[9]:=
Kauffman[Link[11, NonAlternating, 116]][a, z]

Out[9]=
3 5 7 9 4 6 8 10 2 a 4 a 3 a a 3 5 2 a + 3 a + 3 a + a - ---- - ---- - ---- - -- + 7 a z + 15 a z + z z z z 7 9 4 2 6 2 8 2 10 2 12 2 > 12 a z + 4 a z - 3 a z - 5 a z - 2 a z - a z - a z - 3 3 5 3 7 3 9 3 11 3 6 4 8 4 > 6 a z - 22 a z - 27 a z - 19 a z - 8 a z - a z - 7 a z - 10 4 12 4 5 5 7 5 9 5 11 5 > 4 a z + 2 a z + 14 a z + 31 a z + 28 a z + 11 a z - 4 6 6 6 8 6 10 6 12 6 5 7 7 7 > 3 a z + 7 a z + 22 a z + 11 a z - a z - 7 a z - 10 a z - 9 7 11 7 6 8 8 8 10 8 7 9 9 9 > 7 a z - 4 a z - 6 a z - 11 a z - 5 a z - 2 a z - 2 a z

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

Out[10]=
-4 3 1 3 1 4 3 7 4 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 22 9 20 8 18 8 18 7 16 7 16 6 14 6 q q t q t q t q t q t q t q t 7 7 7 8 7 6 4 7 2 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 14 5 12 5 12 4 10 4 10 3 8 3 8 2 6 2 6 q t q t q t q t q t q t q t q t q t 4 > ---- 4 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n116
L11n115
L11n115 L11n117
L11n117

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, NonAlternating, 116]]]

Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 116]]
Out[4]=	PD[X[6, 1, 7, 2], X[7, 17, 8, 16], X[20, 17, 21, 18], X[18, 13, 19, 14], > X[14, 19, 15, 20], X[4, 21, 1, 22], X[10, 5, 11, 6], X[12, 3, 13, 4], > X[22, 11, 5, 12], X[2, 9, 3, 10], X[15, 9, 16, 8]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 8, -6}, {7, -1, -2, 11, 10, -7, 9, -8, 4, -5, -11, 2, 3, -4, > 5, -3, 6, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 4 7 11 14 14 13 11 6 3 q - ----- + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 2 -28 -26 4 2 -20 -18 3 3 -12 4 -q + --- + q - q + --- - --- - q - q - --- + --- - q + --- + 30 24 22 16 14 10 q q q q q q 3 -6 3 > -- - q + -- 8 4 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 116]][a, z]
Out[8]=	3 5 7 9 -2 a 4 a 3 a a 3 5 7 3 3 5 3 ----- + ---- - ---- + -- - 5 a z + 6 a z - 3 a z - 3 a z + 5 a z - z z z z 9 3 5 5 7 5 > a z + 2 a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 116]][a, z]
Out[9]=	3 5 7 9 4 6 8 10 2 a 4 a 3 a a 3 5 2 a + 3 a + 3 a + a - ---- - ---- - ---- - -- + 7 a z + 15 a z + z z z z 7 9 4 2 6 2 8 2 10 2 12 2 > 12 a z + 4 a z - 3 a z - 5 a z - 2 a z - a z - a z - 3 3 5 3 7 3 9 3 11 3 6 4 8 4 > 6 a z - 22 a z - 27 a z - 19 a z - 8 a z - a z - 7 a z - 10 4 12 4 5 5 7 5 9 5 11 5 > 4 a z + 2 a z + 14 a z + 31 a z + 28 a z + 11 a z - 4 6 6 6 8 6 10 6 12 6 5 7 7 7 > 3 a z + 7 a z + 22 a z + 11 a z - a z - 7 a z - 10 a z - 9 7 11 7 6 8 8 8 10 8 7 9 9 9 > 7 a z - 4 a z - 6 a z - 11 a z - 5 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 3 1 3 1 4 3 7 4 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 22 9 20 8 18 8 18 7 16 7 16 6 14 6 q q t q t q t q t q t q t q t 7 7 7 8 7 6 4 7 2 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 14 5 12 5 12 4 10 4 10 3 8 3 8 2 6 2 6 q t q t q t q t q t q t q t q t q t 4 > ---- 4 q t