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DrawMorseLink

PD Presentation:

X10,1,11,2 X16,8,17,7 X18,12,19,11 X19,3,20,2 X3,12,4,13 X13,21,14,20 X14,5,15,6 X6,9,7,10 X22,16,9,15 X8,18,1,17 X4,22,5,21

Gauss Code:

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

Jones Polynomial:

- q-3/2 + 2q-1/2 - 4q1/2 + 4q3/2 - 6q5/2 + 5q7/2 - 4q9/2 + 3q11/2 - 2q13/2 + q15/2

A2 (sl(3)) Invariant:

q-4 + 2 + q2 + 2q4 + 3q6 + q8 + 2q10 - q12 - q18 - q22

HOMFLY-PT Polynomial:

a-5z-1 + 4a-5z + 4a-5z3 + a-5z5 - 3a-3z-1 - 9a-3z - 12a-3z3 - 6a-3z5 - a-3z7 + 2a-1z-1 + 5a-1z + 4a-1z3 + a-1z5

Kauffman Polynomial:

- 3a-8z2 + 4a-8z4 - a-8z6 + a-7z - 7a-7z3 + 8a-7z5 - 2a-7z7 - a-6 + 5a-6z2 - 9a-6z4 + 8a-6z6 - 2a-6z8 + a-5z-1 - 4a-5z + 5a-5z3 - 3a-5z5 + 3a-5z7 - a-5z9 - 3a-4 + 14a-4z2 - 21a-4z4 + 13a-4z6 - 3a-4z8 + 3a-3z-1 - 11a-3z + 18a-3z3 - 14a-3z5 + 5a-3z7 - a-3z9 - 3a-2 + 8a-2z2 - 10a-2z4 + 4a-2z6 - a-2z8 + 2a-1z-1 - 5a-1z + 5a-1z3 - 3a-1z5 + 2z2 - 2z4 + az - az3

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                 1   j = 12               2 1   j = 10             2 1     j = 8           3 2       j = 6         3 2         j = 4       1 3           j = 2     3 3             j = 0   1 3               j = -2   1                 j = -4 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, 206]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 206]]
Out[4]=	PD[X[10, 1, 11, 2], X[16, 8, 17, 7], X[18, 12, 19, 11], X[19, 3, 20, 2], > X[3, 12, 4, 13], X[13, 21, 14, 20], X[14, 5, 15, 6], X[6, 9, 7, 10], > X[22, 16, 9, 15], X[8, 18, 1, 17], X[4, 22, 5, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 4, -5, -11, 7, -8, 2, -10}, > {8, -1, 3, 5, -6, -7, 9, -2, 10, -3, -4, 6, 11, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(3/2) 2 3/2 5/2 7/2 9/2 11/2 -q + ------- - 4 Sqrt[q] + 4 q - 6 q + 5 q - 4 q + 3 q - Sqrt[q] 13/2 15/2 > 2 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-4 2 4 6 8 10 12 18 22 2 + q + q + 2 q + 3 q + q + 2 q - q - q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 206]][a, z]
Out[8]=	3 3 3 5 5 5 7 1 3 2 4 z 9 z 5 z 4 z 12 z 4 z z 6 z z z ---- - ---- + --- + --- - --- + --- + ---- - ----- + ---- + -- - ---- + -- - -- 5 3 a z 5 3 a 5 3 a 5 3 a 3 a z a z a a a a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 206]][a, z]
Out[9]=	-6 3 3 1 3 2 z 4 z 11 z 5 z 2 -a - -- - -- + ---- + ---- + --- + -- - --- - ---- - --- + a z + 2 z - 4 2 5 3 a z 7 5 3 a a a a z a z a a a 2 2 2 2 3 3 3 3 3 z 5 z 14 z 8 z 7 z 5 z 18 z 5 z 3 4 > ---- + ---- + ----- + ---- - ---- + ---- + ----- + ---- - a z - 2 z + 8 6 4 2 7 5 3 a a a a a a a a 4 4 4 4 5 5 5 5 6 6 4 z 9 z 21 z 10 z 8 z 3 z 14 z 3 z z 8 z > ---- - ---- - ----- - ----- + ---- - ---- - ----- - ---- - -- + ---- + 8 6 4 2 7 5 3 a 8 6 a a a a a a a a a 6 6 7 7 7 8 8 8 9 9 13 z 4 z 2 z 3 z 5 z 2 z 3 z z z z > ----- + ---- - ---- + ---- + ---- - ---- - ---- - -- - -- - -- 4 2 7 5 3 6 4 2 5 3 a a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 2 4 4 2 6 2 6 3 3 + 3 q + ----- + - + ---- + 3 q t + q t + 3 q t + 3 q t + 2 q t + 4 2 t 2 q t q t 8 3 8 4 10 4 10 5 12 5 12 6 14 6 16 7 > 3 q t + 2 q t + 2 q t + q t + 2 q t + q t + q t + q t

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