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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X4,15,1,16 X12,6,13,5 X8493 X16,10,17,9 X20,12,21,11 X22,18,5,17 X18,22,19,21 X10,20,11,19 X2,14,3,13

Gauss Code:

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

Jones Polynomial:

q-3/2 - 4q-1/2 + 8q1/2 - 15q3/2 + 18q5/2 - 22q7/2 + 22q9/2 - 19q11/2 + 14q13/2 - 8q15/2 + 4q17/2 - q19/2

A2 (sl(3)) Invariant:

- q-4 + 2q-2 - 1 + q2 + 5q4 - q6 + 7q8 + q12 + 2q14 - 4q16 + 3q18 - 4q20 - q22 + q24 - 2q26 + q28

HOMFLY-PT Polynomial:

a-7z-1 - 2a-7z3 - a-7z5 - 2a-5z-1 - 2a-5z + 2a-5z3 + 3a-5z5 + a-5z7 + 2a-3z + 3a-3z3 + 3a-3z5 + a-3z7 + a-1z-1 - 2a-1z3 - a-1z5

Kauffman Polynomial:

a-11z3 - a-11z5 - 2a-10z2 + 6a-10z4 - 4a-10z6 - 4a-9z3 + 10a-9z5 - 7a-9z7 + 2a-8 - 5a-8z2 + 9a-8z6 - 8a-8z8 - a-7z-1 - 4a-7z3 + 6a-7z5 + 2a-7z7 - 6a-7z9 + 5a-6 - 9a-6z2 - 6a-6z4 + 22a-6z6 - 11a-6z8 - 2a-6z10 - 2a-5z-1 + 3a-5z - 3a-5z3 - a-5z5 + 15a-5z7 - 11a-5z9 + 3a-4 - 7a-4z2 - 4a-4z4 + 21a-4z6 - 9a-4z8 - 2a-4z10 + 4a-3z - 12a-3z3 + 14a-3z5 + 2a-3z7 - 5a-3z9 - a-2 - 2a-2z2 - 2a-2z4 + 11a-2z6 - 6a-2z8 + a-1z-1 + a-1z - 8a-1z3 + 10a-1z5 - 4a-1z7 - z2 + 2z4 - z6

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 20                       1 j = 18                     3   j = 16                   5 1   j = 14                 9 3     j = 12               10 5       j = 10             12 9         j = 8           10 10           j = 6         8 12             j = 4       7 10               j = 2     3 10                 j = 0   1 5                   j = -2   3                     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, Alternating, 6]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 6]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[12, 6, 13, 5], > X[8, 4, 9, 3], X[16, 10, 17, 9], X[20, 12, 21, 11], X[22, 18, 5, 17], > X[18, 22, 19, 21], X[10, 20, 11, 19], X[2, 14, 3, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {4, -1, 2, -5, 6, -10, 7, -4, 11, -2, 3, -6, 8, -9, > 10, -7, 9, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(3/2) 4 3/2 5/2 7/2 9/2 q - ------- + 8 Sqrt[q] - 15 q + 18 q - 22 q + 22 q - Sqrt[q] 11/2 13/2 15/2 17/2 19/2 > 19 q + 14 q - 8 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-4 2 2 4 6 8 12 14 16 18 20 -1 - q + -- + q + 5 q - q + 7 q + q + 2 q - 4 q + 3 q - 4 q - 2 q 22 24 26 28 > q + q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 6]][a, z]
Out[8]=	3 3 3 3 5 5 5 1 2 1 2 z 2 z 2 z 2 z 3 z 2 z z 3 z 3 z ---- - ---- + --- - --- + --- - ---- + ---- + ---- - ---- - -- + ---- + ---- - 7 5 a z 5 3 7 5 3 a 7 5 3 a z a z a a a a a a a a 5 7 7 z z z > -- + -- + -- a 5 3 a a
In[9]:=	Kauffman[Link[11, Alternating, 6]][a, z]
Out[9]=	2 2 2 5 3 -2 1 2 1 3 z 4 z z 2 2 z 5 z -- + -- + -- - a - ---- - ---- + --- + --- + --- + - - z - ---- - ---- - 8 6 4 7 5 a z 5 3 a 10 8 a a a a z a z a a a a 2 2 2 3 3 3 3 3 3 9 z 7 z 2 z z 4 z 4 z 3 z 12 z 8 z 4 > ---- - ---- - ---- + --- - ---- - ---- - ---- - ----- - ---- + 2 z + 6 4 2 11 9 7 5 3 a a a a a a a a a 4 4 4 4 5 5 5 5 5 5 6 z 6 z 4 z 2 z z 10 z 6 z z 14 z 10 z 6 > ---- - ---- - ---- - ---- - --- + ----- + ---- - -- + ----- + ----- - z - 10 6 4 2 11 9 7 5 3 a a a a a a a a a a 6 6 6 6 6 7 7 7 7 7 4 z 9 z 22 z 21 z 11 z 7 z 2 z 15 z 2 z 4 z > ---- + ---- + ----- + ----- + ----- - ---- + ---- + ----- + ---- - ---- - 10 8 6 4 2 9 7 5 3 a a a a a a a a a a 8 8 8 8 9 9 9 10 10 8 z 11 z 9 z 6 z 6 z 11 z 5 z 2 z 2 z > ---- - ----- - ---- - ---- - ---- - ----- - ---- - ----- - ----- 8 6 4 2 7 5 3 6 4 a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 -2 3 5 3 q 4 6 6 2 10 q + 7 q + ----- + t + ----- + - + ---- + 10 q t + 8 q t + 12 q t + 4 3 2 2 t t q t q t 8 2 8 3 10 3 10 4 12 4 12 5 > 10 q t + 10 q t + 12 q t + 9 q t + 10 q t + 5 q t + 14 5 14 6 16 6 16 7 18 7 20 8 > 9 q t + 3 q t + 5 q t + q t + 3 q t + q t

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