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DrawMorseLink

PD Presentation:

X10,1,11,2 X12,4,13,3 X20,5,21,6 X6,9,7,10 X18,12,19,11 X16,8,17,7 X4,14,5,13 X22,16,9,15 X8,18,1,17 X2,19,3,20 X14,22,15,21

Gauss Code:

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

Jones Polynomial:

- q-5/2 + 4q-3/2 - 9q-1/2 + 13q1/2 - 19q3/2 + 21q5/2 - 22q7/2 + 19q9/2 - 14q11/2 + 9q13/2 - 4q15/2 + q17/2

A2 (sl(3)) Invariant:

q-6 - 2q-4 + 3q-2 - 1 + 3q2 + 4q4 - q6 + 7q8 - 4q10 + 3q12 - 2q14 - 2q16 + 2q18 - 3q20 + 2q22 - q24

HOMFLY-PT Polynomial:

a-5z-1 + 3a-5z + 5a-5z3 + 4a-5z5 + a-5z7 - 3a-3z-1 - 7a-3z - 13a-3z3 - 13a-3z5 - 6a-3z7 - a-3z9 + 2a-1z-1 + 4a-1z + 5a-1z3 + 4a-1z5 + a-1z7

Kauffman Polynomial:

- a-10z4 + a-9z3 - 4a-9z5 - 3a-8z2 + 8a-8z4 - 9a-8z6 + a-7z - 8a-7z3 + 17a-7z5 - 13a-7z7 - a-6 + 3a-6z2 - 6a-6z4 + 18a-6z6 - 13a-6z8 + a-5z-1 - 2a-5z - 3a-5z3 + 9a-5z5 + 8a-5z7 - 9a-5z9 - 3a-4 + 13a-4z2 - 36a-4z4 + 45a-4z6 - 12a-4z8 - 3a-4z10 + 3a-3z-1 - 7a-3z + 10a-3z3 - 28a-3z5 + 40a-3z7 - 15a-3z9 - 3a-2 + 10a-2z2 - 33a-2z4 + 31a-2z6 - 3a-2z8 - 3a-2z10 + 2a-1z-1 - 3a-1z + a-1z3 - 13a-1z5 + 18a-1z7 - 6a-1z9 + 3z2 - 12z4 + 13z6 - 4z8 + az - 3az3 + 3az5 - az7

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 18                       1 j = 16                     3   j = 14                   6 1   j = 12                 8 3     j = 10               11 6       j = 8             11 8         j = 6           10 11           j = 4         9 11             j = 2       6 12               j = 0     3 7                 j = -2   1 6                   j = -4   3                     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, Alternating, 330]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 330]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 4, 13, 3], X[20, 5, 21, 6], X[6, 9, 7, 10], > X[18, 12, 19, 11], X[16, 8, 17, 7], X[4, 14, 5, 13], X[22, 16, 9, 15], > X[8, 18, 1, 17], X[2, 19, 3, 20], X[14, 22, 15, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -7, 3, -4, 6, -9}, > {4, -1, 5, -2, 7, -11, 8, -6, 9, -5, 10, -3, 11, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 4 9 3/2 5/2 7/2 -q + ---- - ------- + 13 Sqrt[q] - 19 q + 21 q - 22 q + 3/2 Sqrt[q] q 9/2 11/2 13/2 15/2 17/2 > 19 q - 14 q + 9 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 2 3 2 4 6 8 10 12 14 16 -1 + q - -- + -- + 3 q + 4 q - q + 7 q - 4 q + 3 q - 2 q - 2 q + 4 2 q q 18 20 22 24 > 2 q - 3 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 330]][a, z]
Out[8]=	3 3 3 5 5 1 3 2 3 z 7 z 4 z 5 z 13 z 5 z 4 z 13 z ---- - ---- + --- + --- - --- + --- + ---- - ----- + ---- + ---- - ----- + 5 3 a z 5 3 a 5 3 a 5 3 a z a z a a a a a a 5 7 7 7 9 4 z z 6 z z z > ---- + -- - ---- + -- - -- a 5 3 a 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 330]][a, z]
Out[9]=	2 -6 3 3 1 3 2 z 2 z 7 z 3 z 2 3 z -a - -- - -- + ---- + ---- + --- + -- - --- - --- - --- + a z + 3 z - ---- + 4 2 5 3 a z 7 5 3 a 8 a a a z a z a a a a 2 2 2 3 3 3 3 3 3 z 13 z 10 z z 8 z 3 z 10 z z 3 4 > ---- + ----- + ----- + -- - ---- - ---- + ----- + -- - 3 a z - 12 z - 6 4 2 9 7 5 3 a a a a a a a a 4 4 4 4 4 5 5 5 5 5 z 8 z 6 z 36 z 33 z 4 z 17 z 9 z 28 z 13 z > --- + ---- - ---- - ----- - ----- - ---- + ----- + ---- - ----- - ----- + 10 8 6 4 2 9 7 5 3 a a a a a a a a a a 6 6 6 6 7 7 7 5 6 9 z 18 z 45 z 31 z 13 z 8 z 40 z > 3 a z + 13 z - ---- + ----- + ----- + ----- - ----- + ---- + ----- + 8 6 4 2 7 5 3 a a a a a a a 7 8 8 8 9 9 9 10 18 z 7 8 13 z 12 z 3 z 9 z 15 z 6 z 3 z > ----- - a z - 4 z - ----- - ----- - ---- - ---- - ----- - ---- - ----- - a 6 4 2 5 3 a 4 a a a a a a 10 3 z > ----- 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 3 1 3 6 7 6 q 4 12 q + 9 q + ----- + ----- + ----- + -- + ----- + - + ---- + 11 q t + 6 4 4 3 2 3 2 2 2 t t q t q t q t t q t 6 6 2 8 2 8 3 10 3 10 4 12 4 > 10 q t + 11 q t + 11 q t + 8 q t + 11 q t + 6 q t + 8 q t + 12 5 14 5 14 6 16 6 18 7 > 3 q t + 6 q t + q t + 3 q t + q t

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