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L11n444 L11n446

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DrawMorseLink

PD Presentation:

X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X11,18,12,19 X19,17,20,22 X21,9,22,16 X15,21,16,20 X17,12,18,13 X2536 X9,1,10,4

Gauss Code:

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

Jones Polynomial:

- 2q-3/2 + 3q-1/2 - 9q1/2 + 9q3/2 - 12q5/2 + 9q7/2 - 10q9/2 + 6q11/2 - 3q13/2 + q15/2

A2 (sl(3)) Invariant:

2q-6 + 3q-4 + 3q-2 + 10 + 8q2 + 10q4 + 12q6 + 9q8 + 12q10 + 6q12 + 6q14 + 3q16 - 2q18 + q20 - q22 - q24

HOMFLY-PT Polynomial:

a-7z-1 + a-7z - a-5z-3 - 5a-5z-1 - 6a-5z - 3a-5z3 + 3a-3z-3 + 10a-3z-1 + 12a-3z + 7a-3z3 + 2a-3z5 - 3a-1z-3 - 9a-1z-1 - 9a-1z - 4a-1z3 + az-3 + 3az-1 + 2az

Kauffman Polynomial:

a-8 - 3a-8z2 + 3a-8z4 - a-8z6 - 2a-7z-1 + 6a-7z - 9a-7z3 + 9a-7z5 - 3a-7z7 + 6a-6 - 14a-6z2 + 12a-6z4 + 3a-6z6 - 3a-6z8 + a-5z-3 - 11a-5z-1 + 31a-5z - 45a-5z3 + 38a-5z5 - 9a-5z7 - a-5z9 - 3a-4z-2 + 18a-4 - 30a-4z2 + 9a-4z4 + 13a-4z6 - 7a-4z8 + 3a-3z-3 - 18a-3z-1 + 48a-3z - 66a-3z3 + 43a-3z5 - 10a-3z7 - a-3z9 - 6a-2z-2 + 21a-2 - 24a-2z2 + 8a-2z6 - 4a-2z8 + 3a-1z-3 - 14a-1z-1 + 30a-1z - 33a-1z3 + 14a-1z5 - 4a-1z7 - 3z-2 + 9 - 5z2 - z6 + az-3 - 5az-1 + 7az - 3az3

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 16                   1 j = 14                 2   j = 12               4 1   j = 10             6 2     j = 8           5 6       j = 6         7 4         j = 4     1 3 5           j = 2     7 7             j = 0   2 8               j = -2   1                 j = -4 2

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]=	4
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 445]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 445]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 11, 4, 10], X[7, 15, 8, 14], X[13, 5, 14, 8], > X[11, 18, 12, 19], X[19, 17, 20, 22], X[21, 9, 22, 16], X[15, 21, 16, 20], > X[17, 12, 18, 13], X[2, 5, 3, 6], X[9, 1, 10, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -2, 11}, {10, -1, -3, 4}, {-9, 5, -6, 8, -7, 6}, > {-11, 2, -5, 9, -4, 3, -8, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-2 3 3/2 5/2 7/2 9/2 11/2 ---- + ------- - 9 Sqrt[q] + 9 q - 12 q + 9 q - 10 q + 6 q - 3/2 Sqrt[q] q 13/2 15/2 > 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 3 3 2 4 6 8 10 12 14 10 + -- + -- + -- + 8 q + 10 q + 12 q + 9 q + 12 q + 6 q + 6 q + 6 4 2 q q q 16 18 20 22 24 > 3 q - 2 q + q - q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 445]][a, z]
Out[8]=	1 3 3 a 1 5 10 9 3 a z 6 z -(-----) + ----- - ---- + -- + ---- - ---- + ---- - --- + --- + -- - --- + 5 3 3 3 3 3 7 5 3 a z z 7 5 a z a z a z z a z a z a z a a 3 3 3 5 12 z 9 z 3 z 7 z 4 z 2 z > ---- - --- + 2 a z - ---- + ---- - ---- + ---- 3 a 5 3 a 3 a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 445]][a, z]
Out[9]=	-8 6 18 21 1 3 3 a 3 3 6 9 + a + -- + -- + -- + ----- + ----- + ---- + -- - -- - ----- - ----- - 6 4 2 5 3 3 3 3 3 2 4 2 2 2 a a a a z a z a z z z a z a z 2 11 18 14 5 a 6 z 31 z 48 z 30 z 2 > ---- - ---- - ---- - --- - --- + --- + ---- + ---- + ---- + 7 a z - 5 z - 7 5 3 a z z 7 5 3 a a z a z a z a a a 2 2 2 2 3 3 3 3 3 z 14 z 30 z 24 z 9 z 45 z 66 z 33 z 3 > ---- - ----- - ----- - ----- - ---- - ----- - ----- - ----- - 3 a z + 8 6 4 2 7 5 3 a a a a a a a a 4 4 4 5 5 5 5 6 6 3 z 12 z 9 z 9 z 38 z 43 z 14 z 6 z 3 z > ---- + ----- + ---- + ---- + ----- + ----- + ----- - z - -- + ---- + 8 6 4 7 5 3 a 8 6 a a a a a a a a 6 6 7 7 7 7 8 8 8 9 9 13 z 8 z 3 z 9 z 10 z 4 z 3 z 7 z 4 z z z > ----- + ---- - ---- - ---- - ----- - ---- - ---- - ---- - ---- - -- - -- 4 2 7 5 3 a 6 4 2 5 3 a a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 4 2 2 1 2 4 4 2 6 2 8 + 7 q + q + ----- + - + ---- + 7 q t + 3 q t + 5 q t + 7 q t + 4 2 t 2 q t q t 6 3 8 3 8 4 10 4 10 5 12 5 12 6 > 4 q t + 5 q t + 6 q t + 6 q t + 2 q t + 4 q t + q t + 14 6 16 7 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n445

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