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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-3/2 - 3q-1/2 + 6q1/2 - 11q3/2 + 13q5/2 - 16q7/2 + 16q9/2 - 14q11/2 + 10q13/2 - 6q15/2 + 3q17/2 - q19/2

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- a-11z + 2a-11z3 - a-11z5 - 3a-10z2 + 6a-10z4 - 3a-10z6 - a-9z + 5a-9z5 - 4a-9z7 + a-8z4 + 3a-8z6 - 4a-8z8 - 3a-7z + 4a-7z3 - a-7z5 + a-7z7 - 3a-7z9 + 3a-6z2 - 11a-6z4 + 13a-6z6 - 6a-6z8 - a-6z10 - a-5z-1 + 3a-5z3 - 7a-5z5 + 10a-5z7 - 6a-5z9 + a-4 + a-4z2 - 12a-4z4 + 17a-4z6 - 6a-4z8 - a-4z10 - a-3z-1 + 6a-3z - 11a-3z3 + 9a-3z5 + 2a-3z7 - 3a-3z9 - a-2z2 - 3a-2z4 + 9a-2z6 - 4a-2z8 + 3a-1z - 8a-1z3 + 9a-1z5 - 3a-1z7 - 2z2 + 3z4 - 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                     2   j = 16                   4 1   j = 14                 6 2     j = 12               8 4       j = 10             9 7         j = 8           7 7           j = 6         6 9             j = 4       5 7               j = 2     2 7                 j = 0   1 4                   j = -2   2                     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, 284]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 284]]
Out[4]=	PD[X[10, 1, 11, 2], X[22, 11, 9, 12], X[8, 9, 1, 10], X[2, 22, 3, 21], > X[14, 6, 15, 5], X[12, 4, 13, 3], X[4, 14, 5, 13], X[16, 20, 17, 19], > X[18, 8, 19, 7], X[6, 18, 7, 17], X[20, 16, 21, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 6, -7, 5, -10, 9, -3}, > {3, -1, 2, -6, 7, -5, 11, -8, 10, -9, 8, -11, 4, -2}]
In[6]:=	Jones[L][q]
Out[6]=	-(3/2) 3 3/2 5/2 7/2 9/2 q - ------- + 6 Sqrt[q] - 11 q + 13 q - 16 q + 16 q - Sqrt[q] 11/2 13/2 15/2 17/2 19/2 > 14 q + 10 q - 6 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-4 -2 4 6 8 12 14 16 18 20 24 -1 - q + q + 3 q - q + 5 q + q + 2 q - 2 q + 3 q - 2 q + q - 26 28 > q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 284]][a, z]
Out[8]=	3 3 3 3 5 5 1 1 2 z z 5 z 2 z 3 z 5 z 6 z 3 z z 4 z -(----) + ---- - --- + -- + --- - --- - ---- + ---- + ---- - ---- - -- + ---- + 5 3 7 5 3 a 7 5 3 a 7 5 a z a z a a a a a a a a 5 5 7 7 4 z z z z > ---- - -- + -- + -- 3 a 5 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 284]][a, z]
Out[9]=	2 2 2 2 -4 1 1 z z 3 z 6 z 3 z 2 3 z 3 z z z a - ---- - ---- - --- - -- - --- + --- + --- - 2 z - ---- + ---- + -- - -- + 5 3 11 9 7 3 a 10 6 4 2 a z a z a a a a a a a a 3 3 3 3 3 4 4 4 4 2 z 4 z 3 z 11 z 8 z 4 6 z z 11 z 12 z > ---- + ---- + ---- - ----- - ---- + 3 z + ---- + -- - ----- - ----- - 11 7 5 3 a 10 8 6 4 a a a a a a a a 4 5 5 5 5 5 5 6 6 6 3 z z 5 z z 7 z 9 z 9 z 6 3 z 3 z 13 z > ---- - --- + ---- - -- - ---- + ---- + ---- - z - ---- + ---- + ----- + 2 11 9 7 5 3 a 10 8 6 a a a a a a a a a 6 6 7 7 7 7 7 8 8 8 17 z 9 z 4 z z 10 z 2 z 3 z 4 z 6 z 6 z > ----- + ---- - ---- + -- + ----- + ---- - ---- - ---- - ---- - ---- - 4 2 9 7 5 3 a 8 6 4 a a a a a a a a a 8 9 9 9 10 10 4 z 3 z 6 z 3 z z z > ---- - ---- - ---- - ---- - --- - --- 2 7 5 3 6 4 a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 -2 2 4 2 q 4 6 6 2 7 q + 5 q + ----- + t + ----- + - + ---- + 7 q t + 6 q t + 9 q t + 4 3 2 2 t t q t q t 8 2 8 3 10 3 10 4 12 4 12 5 14 5 > 7 q t + 7 q t + 9 q t + 7 q t + 8 q t + 4 q t + 6 q t + 14 6 16 6 16 7 18 7 20 8 > 2 q t + 4 q t + q t + 2 q t + q t

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