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DrawMorseLink

PD Presentation:

X8192 X11,19,12,18 X3,10,4,11 X17,3,18,2 X12,5,13,6 X6718 X16,10,17,9 X20,16,21,15 X22,14,7,13 X14,22,15,21 X4,20,5,19

Gauss Code:

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

Jones Polynomial:

- q-5/2 + q-3/2 - 2q-1/2 + 2q1/2 - 3q3/2 + 2q5/2 - 3q7/2 + 2q9/2 - q11/2 + q13/2

A2 (sl(3)) Invariant:

q-8 + q-6 + q-4 + q-2 + 1 + 2q2 + 2q4 + 2q6 + 2q8 - q14 - q16 - q18 - q20

HOMFLY-PT Polynomial:

a-5z-1 + 3a-5z + a-5z3 - 2a-3z-1 - 4a-3z - 4a-3z3 - a-3z5 - 3a-1z - 4a-1z3 - a-1z5 + az-1 + 3az + az3

Kauffman Polynomial:

- 2a-6 + 13a-6z2 - 16a-6z4 + 7a-6z6 - a-6z8 + a-5z-1 - 4a-5z + 8a-5z3 - 11a-5z5 + 6a-5z7 - a-5z9 - 5a-4 + 22a-4z2 - 33a-4z4 + 18a-4z6 - 3a-4z8 + 2a-3z-1 - 5a-3z - 2a-3z5 + 4a-3z7 - a-3z9 - 3a-2 + 9a-2z2 - 14a-2z4 + 10a-2z6 - 2a-2z8 + 4a-1z - 10a-1z3 + 9a-1z5 - 2a-1z7 + 1 - z2 + 3z4 - z6 - az-1 + 4az - 2az3 - a2z2 - a3z

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 14                   1 j = 12                     j = 10               2 1   j = 8             1       j = 6           1 2       j = 4         2 1         j = 2         1           j = 0     2 2             j = -2     1               j = -4 1 1                 j = -6 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, 143]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 143]]
Out[4]=	PD[X[8, 1, 9, 2], X[11, 19, 12, 18], X[3, 10, 4, 11], X[17, 3, 18, 2], > X[12, 5, 13, 6], X[6, 7, 1, 8], X[16, 10, 17, 9], X[20, 16, 21, 15], > X[22, 14, 7, 13], X[14, 22, 15, 21], X[4, 20, 5, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 4, -3, -11, 5, -6}, > {6, -1, 7, 3, -2, -5, 9, -10, 8, -7, -4, 2, 11, -8, 10, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) -(3/2) 2 3/2 5/2 7/2 9/2 -q + q - ------- + 2 Sqrt[q] - 3 q + 2 q - 3 q + 2 q - Sqrt[q] 11/2 13/2 > q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 -6 -4 -2 2 4 6 8 14 16 18 20 1 + q + q + q + q + 2 q + 2 q + 2 q + 2 q - q - q - q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 143]][a, z]
Out[8]=	3 3 3 5 5 1 2 a 3 z 4 z 3 z z 4 z 4 z 3 z z ---- - ---- + - + --- - --- - --- + 3 a z + -- - ---- - ---- + a z - -- - -- 5 3 z 5 3 a 5 3 a 3 a a z a z a a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 143]][a, z]
Out[9]=	2 5 3 1 2 a 4 z 5 z 4 z 3 2 1 - -- - -- - -- + ---- + ---- - - - --- - --- + --- + 4 a z - a z - z + 6 4 2 5 3 z 5 3 a a a a a z a z a a 2 2 2 3 3 4 13 z 22 z 9 z 2 2 8 z 10 z 3 4 16 z > ----- + ----- + ---- - a z + ---- - ----- - 2 a z + 3 z - ----- - 6 4 2 5 a 6 a a a a a 4 4 5 5 5 6 6 6 7 33 z 14 z 11 z 2 z 9 z 6 7 z 18 z 10 z 6 z > ----- - ----- - ----- - ---- + ---- - z + ---- + ----- + ----- + ---- + 4 2 5 3 a 6 4 2 5 a a a a a a a a 7 7 8 8 8 9 9 4 z 2 z z 3 z 2 z z z > ---- - ---- - -- - ---- - ---- - -- - -- 3 a 6 4 2 5 3 a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	-2 1 1 1 2 2 4 2 4 3 6 3 2 + q + ----- + ----- + ---- + 2 t + q t + 2 q t + q t + q t + 6 2 4 2 4 q t q t q t 6 4 8 4 10 5 10 6 14 7 > 2 q t + q t + 2 q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n143

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