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L11n119 L11n121

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DrawMorseLink

PD Presentation:

X6172 X14,3,15,4 X9,16,10,17 X11,21,12,20 X21,9,22,8 X7,19,8,18 X19,13,20,12 X15,10,16,11 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-11/2 + 2q-9/2 - 3q-7/2 + 2q-5/2 - 2q-3/2 - q3/2 + 2q5/2 - 2q7/2 + q9/2

A2 (sl(3)) Invariant:

q-18 + q-16 + q-12 + 2q-8 + q-6 + q-4 + 2q-2 + 2q2 - q10 - q14

HOMFLY-PT Polynomial:

a-3z-1 + 2a-3z + a-3z3 - 3a-1z-1 - 7a-1z - 5a-1z3 - a-1z5 + 3az-1 + 7az + 5az3 + az5 - 2a3z-1 - 5a3z - 2a3z3 + a5z-1 + a5z

Kauffman Polynomial:

a-4 - 3a-4z2 + 4a-4z4 - a-4z6 - a-3z-1 + 3a-3z - 8a-3z3 + 9a-3z5 - 2a-3z7 + 2a-2 - 6a-2z2 + a-2z4 + 4a-2z6 - a-2z8 - 3a-1z-1 + 13a-1z - 22a-1z3 + 14a-1z5 - 2a-1z7 + 6z2 - 18z4 + 13z6 - 2z8 - 3az-1 + 16az - 23az3 + 6az5 + 4az7 - az9 - 2a2 + 13a2z2 - 27a2z4 + 18a2z6 - 3a2z8 - 2a3z-1 + 10a3z - 16a3z3 + 6a3z5 + 3a3z7 - a3z9 + 4a4z2 - 12a4z4 + 10a4z6 - 2a4z8 - a5z-1 + 4a5z - 7a5z3 + 5a5z5 - a5z7

Khovanov Homology:

trqj 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 = 10                       1 j = 8                     1   j = 6                   1 1   j = 4               2 2 1     j = 2             1 1 1       j = 0           2 4 2         j = -2         2 2 2           j = -4       1 2 1             j = -6     2 2 1               j = -8   1 2                   j = -10   1                     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, NonAlternating, 120]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 120]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[9, 16, 10, 17], X[11, 21, 12, 20], > X[21, 9, 22, 8], X[7, 19, 8, 18], X[19, 13, 20, 12], X[15, 10, 16, 11], > 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]=	-(11/2) 2 3 2 2 3/2 5/2 7/2 9/2 -q + ---- - ---- + ---- - ---- - q + 2 q - 2 q + q 9/2 7/2 5/2 3/2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 -12 2 -6 -4 2 2 10 14 q + q + q + -- + q + q + -- + 2 q - q - q 8 2 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 120]][a, z]
Out[8]=	3 5 3 3 1 3 3 a 2 a a 2 z 7 z 3 5 z 5 z ---- - --- + --- - ---- + -- + --- - --- + 7 a z - 5 a z + a z + -- - ---- + 3 a z z z z 3 a 3 a a z a a 5 3 3 3 z 5 > 5 a z - 2 a z - -- + a z a
In[9]:=	Kauffman[Link[11, NonAlternating, 120]][a, z]
Out[9]=	3 5 -4 2 2 1 3 3 a 2 a a 3 z 13 z a + -- - 2 a - ---- - --- - --- - ---- - -- + --- + ---- + 16 a z + 2 3 a z z z z 3 a a a z a 2 2 3 3 3 5 2 3 z 6 z 2 2 4 2 8 z 22 z > 10 a z + 4 a z + 6 z - ---- - ---- + 13 a z + 4 a z - ---- - ----- - 4 2 3 a a a a 4 4 3 3 3 5 3 4 4 z z 2 4 4 4 > 23 a z - 16 a z - 7 a z - 18 z + ---- + -- - 27 a z - 12 a z + 4 2 a a 5 5 6 6 9 z 14 z 5 3 5 5 5 6 z 4 z 2 6 > ---- + ----- + 6 a z + 6 a z + 5 a z + 13 z - -- + ---- + 18 a z + 3 a 4 2 a a a 7 7 8 4 6 2 z 2 z 7 3 7 5 7 8 z 2 8 > 10 a z - ---- - ---- + 4 a z + 3 a z - a z - 2 z - -- - 3 a z - 3 a 2 a a 4 8 9 3 9 > 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 1 1 2 2 2 1 1 4 + -- + q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 2 12 6 10 5 8 5 8 4 6 4 6 3 4 3 6 2 q q t q t q t q t q t q t q t q t 2 2 2 1 2 2 4 2 2 4 2 > ----- + ----- + - + ---- + ---- + 2 t + q t + 2 q t + q t + 2 q t + 4 2 2 2 t 4 2 q t q t q t q t 4 3 6 3 6 4 8 4 10 5 > q t + q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n120

L11n119 L11n121