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DrawMorseLink

PD Presentation:

X8192 X9,21,10,20 X6,21,1,22 X18,8,19,7 X3,10,4,11 X15,12,16,13 X5,14,6,15 X13,4,14,5 X11,16,12,17 X22,18,7,17 X19,2,20,3

Gauss Code:

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

Jones Polynomial:

- q-17/2 + 2q-15/2 - 3q-13/2 + 4q-11/2 - 5q-9/2 + 4q-7/2 - 5q-5/2 + 3q-3/2 - 2q-1/2 + q1/2

A2 (sl(3)) Invariant:

q-26 - q-18 + 2q-16 + 2q-14 + 3q-12 + 3q-10 + q-8 + q-6 - q-4 - q-2 - q2

HOMFLY-PT Polynomial:

az-1 + 2az + az3 - 3a3z-1 - 6a3z - 4a3z3 - a3z5 + 2a5z-1 - a5z - 3a5z3 - a5z5 + 2a7z + a7z3

Kauffman Polynomial:

1 - z2 - az-1 + 2az - 2az3 + 3a2 - 6a2z2 + 3a2z4 - a2z6 - 3a3z-1 + 10a3z - 13a3z3 + 8a3z5 - 2a3z7 + 3a4 - 2a4z2 - 5a4z4 + 7a4z6 - 2a4z8 - 2a5z-1 + 7a5z - 14a5z3 + 9a5z5 + a5z7 - a5z9 + 9a6z2 - 22a6z4 + 18a6z6 - 4a6z8 + 2a7z - 10a7z3 + 6a7z5 + 2a7z7 - a7z9 + 6a8z2 - 14a8z4 + 10a8z6 - 2a8z8 + 3a9z - 7a9z3 + 5a9z5 - a9z7

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 j = 2                   1 j = 0                 1   j = -2               3 2   j = -4             2       j = -6           2 3       j = -8         3 2         j = -10       1 2           j = -12     2 3             j = -14   1 2               j = -16   1                 j = -18 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, 184]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 184]]
Out[4]=	PD[X[8, 1, 9, 2], X[9, 21, 10, 20], X[6, 21, 1, 22], X[18, 8, 19, 7], > X[3, 10, 4, 11], X[15, 12, 16, 13], X[5, 14, 6, 15], X[13, 4, 14, 5], > X[11, 16, 12, 17], X[22, 18, 7, 17], X[19, 2, 20, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 11, -5, 8, -7, -3}, > {4, -1, -2, 5, -9, 6, -8, 7, -6, 9, 10, -4, -11, 2, 3, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 2 3 4 5 4 5 3 2 -q + ----- - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q > Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 -18 2 2 3 3 -8 -6 -4 -2 2 q - q + --- + --- + --- + --- + q + q - q - q - q 16 14 12 10 q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 184]][a, z]
Out[8]=	3 5 a 3 a 2 a 3 5 7 3 3 3 5 3 - - ---- + ---- + 2 a z - 6 a z - a z + 2 a z + a z - 4 a z - 3 a z + z z z 7 3 3 5 5 5 > a z - a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 184]][a, z]
Out[9]=	3 5 2 4 a 3 a 2 a 3 5 7 1 + 3 a + 3 a - - - ---- - ---- + 2 a z + 10 a z + 7 a z + 2 a z + z z z 9 2 2 2 4 2 6 2 8 2 3 3 3 > 3 a z - z - 6 a z - 2 a z + 9 a z + 6 a z - 2 a z - 13 a z - 5 3 7 3 9 3 2 4 4 4 6 4 8 4 > 14 a z - 10 a z - 7 a z + 3 a z - 5 a z - 22 a z - 14 a z + 3 5 5 5 7 5 9 5 2 6 4 6 6 6 > 8 a z + 9 a z + 6 a z + 5 a z - a z + 7 a z + 18 a z + 8 6 3 7 5 7 7 7 9 7 4 8 6 8 > 10 a z - 2 a z + a z + 2 a z - a z - 2 a z - 4 a z - 8 8 5 9 7 9 > 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 2 2 3 1 1 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 18 8 16 7 14 7 14 6 12 6 12 5 10 5 q q t q t q t q t q t q t q t 2 3 2 2 3 2 3 2 > ------ + ----- + ----- + ----- + ----- + ----- + ---- + q t 10 4 8 4 8 3 6 3 6 2 4 2 2 q t q t q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n184

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