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L11n196 L11n198

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DrawMorseLink

PD Presentation:

X8192 X12,3,13,4 X22,10,7,9 X10,14,11,13 X5,19,6,18 X21,16,22,17 X15,20,16,21 X19,14,20,15 X2738 X4,11,5,12 X17,1,18,6

Gauss Code:

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

Jones Polynomial:

q-15/2 - q-13/2 + q-11/2 - q-9/2 - q-7/2 - 2q-3/2 + 2q-1/2 - 2q1/2 + 2q3/2 - q5/2

A2 (sl(3)) Invariant:

- q-24 - q-22 - q-20 - q-18 + q-16 + q-14 + 3q-12 + 3q-10 + 3q-8 + 3q-6 - 1 - q2 + q8

HOMFLY-PT Polynomial:

- 2a-1z - a-1z3 + 3az + 4az3 + az5 - 2a3z-1 - 4a3z - a3z3 + 3a5z-1 + 3a5z + a5z3 - a7z-1 - a7z

Kauffman Polynomial:

2a-1z - 6a-1z3 + 5a-1z5 - a-1z7 + 7z2 - 16z4 + 11z6 - 2z8 - 5az3 + 4az7 - az9 + 12a2z2 - 28a2z4 + 18a2z6 - 3a2z8 - 2a3z-1 - a3z + 6a3z3 - 9a3z5 + 6a3z7 - a3z9 + 3a4 + 4a4z2 - 11a4z4 + 7a4z6 - a4z8 - 3a5z-1 + 4a5z - a5z3 + a5z5 + 3a6 - 7a6z2 + 6a6z4 - a6z6 - a7z-1 + 3a7z - 6a7z3 + 5a7z5 - a7z7 + a8 - 6a8z2 + 5a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 6                       1 j = 4                     1   j = 2                   1 1   j = 0               1 2 1     j = -2             1 2 1       j = -4           1 1 2         j = -6         1 3 1           j = -8       1   1             j = -10       1 1               j = -12   1 1                   j = -14                         j = -16 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, 197]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 197]]
Out[4]=	PD[X[8, 1, 9, 2], X[12, 3, 13, 4], X[22, 10, 7, 9], X[10, 14, 11, 13], > X[5, 19, 6, 18], X[21, 16, 22, 17], X[15, 20, 16, 21], X[19, 14, 20, 15], > X[2, 7, 3, 8], X[4, 11, 5, 12], X[17, 1, 18, 6]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10, -5, 11}, > {9, -1, 3, -4, 10, -2, 4, 8, -7, 6, -11, 5, -8, 7, -6, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) -(13/2) -(11/2) -(9/2) -(7/2) 2 2 q - q + q - q - q - ---- + ------- - 3/2 Sqrt[q] q 3/2 5/2 > 2 Sqrt[q] + 2 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -22 -20 -18 -16 -14 3 3 3 3 2 8 -1 - q - q - q - q + q + q + --- + --- + -- + -- - q + q 12 10 8 6 q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 197]][a, z]
Out[8]=	3 5 7 3 -2 a 3 a a 2 z 3 5 7 z 3 ----- + ---- - -- - --- + 3 a z - 4 a z + 3 a z - a z - -- + 4 a z - z z z a a 3 3 5 3 5 > a z + a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 197]][a, z]
Out[9]=	3 5 7 4 6 8 2 a 3 a a 2 z 3 5 7 2 3 a + 3 a + a - ---- - ---- - -- + --- - a z + 4 a z + 3 a z + 7 z + z z z a 3 2 2 4 2 6 2 8 2 6 z 3 3 3 5 3 > 12 a z + 4 a z - 7 a z - 6 a z - ---- - 5 a z + 6 a z - a z - a 5 7 3 4 2 4 4 4 6 4 8 4 5 z > 6 a z - 16 z - 28 a z - 11 a z + 6 a z + 5 a z + ---- - a 3 5 5 5 7 5 6 2 6 4 6 6 6 8 6 > 9 a z + a z + 5 a z + 11 z + 18 a z + 7 a z - a z - a z - 7 z 7 3 7 7 7 8 2 8 4 8 9 3 9 > -- + 4 a z + 6 a z - a z - 2 z - 3 a z - a z - a z - a z a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 1 1 1 1 1 1 1 + -- + -- + ------ + ------ + ------ + ------ + ----- + ------ + ----- + 4 2 16 7 12 6 12 5 10 4 8 4 10 3 6 3 q q q t q t q t q t q t q t q t 1 3 1 1 1 1 t 2 2 2 > ----- + ----- + ----- + ---- + ---- + ---- + 2 t + -- + t + q t + 8 2 6 2 4 2 6 4 2 2 q t q t q t q t q t q t q 2 3 4 3 6 4 > q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n197

L11n196 L11n198