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L11n121 L11n123

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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 X22,17,5,18 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-7/2 - 2q-1/2 + q1/2 - 2q3/2 + 3q5/2 - 2q7/2 + 2q9/2 - q11/2

A2 (sl(3)) Invariant:

q-12 + 2q-10 + 3q-8 + 2q-6 + 2q-4 + 2q-2 - q4 - q6 - q10 - q14 + q18

HOMFLY-PT Polynomial:

- a-5z + a-3z + a-3z3 + a-1z-1 + 2a-1z + a-1z3 - 3az-1 - 5az - az3 + 2a3z-1 + a3z

Kauffman Polynomial:

a-5z - 6a-5z3 + 5a-5z5 - a-5z7 + 9a-4z2 - 17a-4z4 + 11a-4z6 - 2a-4z8 - 3a-3z3 - a-3z5 + 4a-3z7 - a-3z9 - a-2 + 13a-2z2 - 26a-2z4 + 17a-2z6 - 3a-2z8 + a-1z-1 - 3a-1z + 2a-1z3 - 5a-1z5 + 5a-1z7 - a-1z9 - 3 + 7z2 - 9z4 + 6z6 - z8 + 3az-1 - 7az + 4az3 - 3a2 + 3a2z2 + 2a3z-1 - 5a3z + 5a3z3 - a3z5

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 = 12                     1 j = 10                   1   j = 8                 1 1   j = 6               2 1     j = 4           1 1 1       j = 2           1 2         j = 0       1 3 1           j = -2         2             j = -4     1 1               j = -6 1                     j = -8 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, 122]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 122]]
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[22, 17, 5, 18], 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]=	-(7/2) 2 3/2 5/2 7/2 9/2 11/2 -q - ------- + Sqrt[q] - 2 q + 3 q - 2 q + 2 q - q Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 2 3 2 2 2 4 6 10 14 18 q + --- + -- + -- + -- + -- - q - q - q - q + q 10 8 6 4 2 q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 122]][a, z]
Out[8]=	3 3 3 1 3 a 2 a z z 2 z 3 z z 3 --- - --- + ---- - -- + -- + --- - 5 a z + a z + -- + -- - a z a z z z 5 3 a 3 a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 122]][a, z]
Out[9]=	3 2 -2 2 1 3 a 2 a z 3 z 3 2 9 z -3 - a - 3 a + --- + --- + ---- + -- - --- - 7 a z - 5 a z + 7 z + ---- + a z z z 5 a 4 a a 2 3 3 3 4 13 z 2 2 6 z 3 z 2 z 3 3 3 4 17 z > ----- + 3 a z - ---- - ---- + ---- + 4 a z + 5 a z - 9 z - ----- - 2 5 3 a 4 a a a a 4 5 5 5 6 6 7 7 26 z 5 z z 5 z 3 5 6 11 z 17 z z 4 z > ----- + ---- - -- - ---- - a z + 6 z + ----- + ----- - -- + ---- + 2 5 3 a 4 2 5 3 a a a a a a a 7 8 8 9 9 5 z 8 2 z 3 z z z > ---- - z - ---- - ---- - -- - -- a 4 2 3 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 1 1 2 4 2 2 4 2 3 + -- + ----- + ----- + ----- + - + ---- + t + q t + q t + 2 q t + q t + 2 8 4 6 4 4 2 t 4 q q t q t q t q t 4 3 6 3 6 4 8 4 8 5 10 5 12 6 > q t + 2 q t + q t + q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n122

L11n121 L11n123