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L11n154 L11n156

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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X7,16,8,17 X13,20,14,21 X15,22,16,7 X6,19,1,20 X18,11,19,12 X12,6,13,5 X21,14,22,15 X4,18,5,17

Gauss Code:

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

Jones Polynomial:

q-23/2 - 3q-21/2 + 6q-19/2 - 8q-17/2 + 10q-15/2 - 10q-13/2 + 8q-11/2 - 7q-9/2 + 3q-7/2 - 2q-5/2

A2 (sl(3)) Invariant:

- q-34 + q-32 - 2q-30 - q-28 - q-26 - 3q-24 + 2q-22 - q-20 + 4q-18 + 3q-16 + 2q-14 + 4q-12 + 2q-8

HOMFLY-PT Polynomial:

- 3a5z-1 - 9a5z - 8a5z3 - 2a5z5 + 5a7z-1 + 11a7z + 10a7z3 + 5a7z5 + a7z7 - 2a9z-1 - 3a9z - 3a9z3 - a9z5

Kauffman Polynomial:

3a5z-1 - 10a5z + 10a5z3 - 3a5z5 - 5a6 + 9a6z2 - 4a6z4 + 2a6z6 - a6z8 + 5a7z-1 - 15a7z + 19a7z3 - 8a7z5 + 2a7z7 - a7z9 - 5a8 + 11a8z2 - 9a8z4 + 8a8z6 - 4a8z8 + 2a9z-1 - 4a9z + a9z3 + 5a9z5 - 3a9z7 - a9z9 - 3a10z2 + 3a10z4 + a10z6 - 3a10z8 + a11z - 5a11z3 + 7a11z5 - 5a11z7 + a12 - 4a12z2 + 7a12z4 - 5a12z6 + 3a13z3 - 3a13z5 + a14z2 - a14z4

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -4                   2 j = -6                 2 1 j = -8               5 1   j = -10             4 3     j = -12           6 4       j = -14         4 4         j = -16       4 6           j = -18     2 4             j = -20   1 4               j = -22   2                 j = -24 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, 155]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 155]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[7, 16, 8, 17], > X[13, 20, 14, 21], X[15, 22, 16, 7], X[6, 19, 1, 20], X[18, 11, 19, 12], > X[12, 6, 13, 5], X[21, 14, 22, 15], X[4, 18, 5, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -11, 9, -7}, > {-4, -1, 2, -3, 8, -9, -5, 10, -6, 4, 11, -8, 7, 5, -10, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-(23/2) 3 6 8 10 10 8 7 3 2 q - ----- + ----- - ----- + ----- - ----- + ----- - ---- + ---- - ---- 21/2 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 -32 2 -28 -26 3 2 -20 4 3 2 4 2 -q + q - --- - q - q - --- + --- - q + --- + --- + --- + --- + -- 30 24 22 18 16 14 12 8 q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 155]][a, z]
Out[8]=	5 7 9 -3 a 5 a 2 a 5 7 9 5 3 7 3 ----- + ---- - ---- - 9 a z + 11 a z - 3 a z - 8 a z + 10 a z - z z z 9 3 5 5 7 5 9 5 7 7 > 3 a z - 2 a z + 5 a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 155]][a, z]
Out[9]=	5 7 9 6 8 12 3 a 5 a 2 a 5 7 9 11 -5 a - 5 a + a + ---- + ---- + ---- - 10 a z - 15 a z - 4 a z + a z + z z z 6 2 8 2 10 2 12 2 14 2 5 3 7 3 > 9 a z + 11 a z - 3 a z - 4 a z + a z + 10 a z + 19 a z + 9 3 11 3 13 3 6 4 8 4 10 4 12 4 > a z - 5 a z + 3 a z - 4 a z - 9 a z + 3 a z + 7 a z - 14 4 5 5 7 5 9 5 11 5 13 5 6 6 > a z - 3 a z - 8 a z + 5 a z + 7 a z - 3 a z + 2 a z + 8 6 10 6 12 6 7 7 9 7 11 7 6 8 > 8 a z + a z - 5 a z + 2 a z - 3 a z - 5 a z - a z - 8 8 10 8 7 9 9 9 > 4 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 2 1 2 1 4 2 4 4 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 24 9 22 8 20 8 20 7 18 7 18 6 16 6 q q t q t q t q t q t q t q t 6 4 4 6 4 4 3 5 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 16 5 14 5 14 4 12 4 12 3 10 3 10 2 8 2 q t q t q t q t q t q t q t q t 1 2 > ---- + ---- 8 6 q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n155

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