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L11n120 L11n122

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X14,3,15,4 X16,10,17,9 X11,21,12,20 X21,9,22,8 X7,19,8,18 X19,13,20,12 X10,16,11,15 X17,5,18,22 X2536 X4,13,1,14

Gauss Code:

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

Jones Polynomial:

- q-5/2 - 3q1/2 + 3q3/2 - 4q5/2 + 4q7/2 - 3q9/2 + 3q11/2 - q13/2

A2 (sl(3)) Invariant:

q-8 + 2q-6 + 3q-4 + 2q-2 + 4 + 2q2 + q4 + q6 - 2q8 - q10 - 3q12 - q14 - q18 + q20

HOMFLY-PT Polynomial:

- a-5z - a-5z3 + 2a-3z-1 + 6a-3z + 4a-3z3 + a-3z5 - 5a-1z-1 - 11a-1z - 7a-1z3 - a-1z5 + 3az-1 + 4az + az3

Kauffman Polynomial:

2a-7z3 - a-7z5 - a-6 - 4a-6z2 + 9a-6z4 - 3a-6z6 + a-5z - 2a-5z3 + 5a-5z5 - 2a-5z7 - 5a-4z2 + 10a-4z4 - 4a-4z6 - 2a-3z-1 + 5a-3z - 4a-3z3 + 2a-3z5 - a-3z7 + 5a-2 - 10a-2z2 + 4a-2z4 - a-2z6 - 5a-1z-1 + 14a-1z - 14a-1z3 + 3a-1z5 + 5 - 9z2 + 3z4 - 3az-1 + 10az - 14az3 + 7az5 - az7

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 14                     1 j = 12                   2   j = 10                 1 1   j = 8               3 2     j = 6           1 2 1       j = 4           2 3         j = 2       1 3 2           j = 0         3             j = -2     1 1               j = -4 1                     j = -6 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, 121]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 121]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[16, 10, 17, 9], X[11, 21, 12, 20], > X[21, 9, 22, 8], X[7, 19, 8, 18], X[19, 13, 20, 12], X[10, 16, 11, 15], > X[17, 5, 18, 22], X[2, 5, 3, 6], X[4, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -6, 5, 3, -8, -4, 7, 11, -2, 8, -3, -9, 6, > -7, 4, -5, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 3/2 5/2 7/2 9/2 11/2 13/2 -q - 3 Sqrt[q] + 3 q - 4 q + 4 q - 3 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 2 3 2 2 4 6 8 10 12 14 18 20 4 + q + -- + -- + -- + 2 q + q + q - 2 q - q - 3 q - q - q + q 6 4 2 q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 121]][a, z]
Out[8]=	3 3 3 5 5 2 5 3 a z 6 z 11 z z 4 z 7 z 3 z z ---- - --- + --- - -- + --- - ---- + 4 a z - -- + ---- - ---- + a z + -- - -- 3 a z z 5 3 a 5 3 a 3 a a z a a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 121]][a, z]
Out[9]=	2 -6 5 2 5 3 a z 5 z 14 z 2 4 z 5 - a + -- - ---- - --- - --- + -- + --- + ---- + 10 a z - 9 z - ---- - 2 3 a z z 5 3 a 6 a a z a a a 2 2 3 3 3 3 4 4 5 z 10 z 2 z 2 z 4 z 14 z 3 4 9 z 10 z > ---- - ----- + ---- - ---- - ---- - ----- - 14 a z + 3 z + ---- + ----- + 4 2 7 5 3 a 6 4 a a a a a a a 4 5 5 5 5 6 6 6 7 7 4 z z 5 z 2 z 3 z 5 3 z 4 z z 2 z z > ---- - -- + ---- + ---- + ---- + 7 a z - ---- - ---- - -- - ---- - -- - 2 7 5 3 a 6 4 2 5 3 a a a a a a a a a 7 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 1 1 1 q 2 4 6 3 + 3 q + ----- + ----- + ----- + ---- + -- + 2 q t + 2 q t + q t + 6 4 4 4 2 2 2 t q t q t q t q t 4 2 6 2 6 3 8 3 8 4 10 4 10 5 > 3 q t + 2 q t + q t + 3 q t + 2 q t + q t + q t + 12 5 14 6 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n121

L11n120 L11n122