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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-1/2 + 3q1/2 - 7q3/2 + 11q5/2 - 16q7/2 + 18q9/2 - 19q11/2 + 16q13/2 - 12q15/2 + 8q17/2 - 4q19/2 + q21/2

A2 (sl(3)) Invariant:

q-2 - 1 + 3q4 - 3q6 + 3q8 + 2q10 - q12 + 4q14 - q16 + 3q18 - 2q22 + 3q24 - 3q26 + 2q30 - q32

HOMFLY-PT Polynomial:

a-9z3 + a-7z - a-7z5 - a-5z-1 - 2a-5z - 3a-5z3 - 2a-5z5 + a-3z-1 + 2a-3z - a-3z5 + a-1z + a-1z3

Kauffman Polynomial:

- a-12z2 + 2a-12z4 - a-12z6 + a-11z - 6a-11z3 + 10a-11z5 - 4a-11z7 - 7a-10z4 + 14a-10z6 - 6a-10z8 + 3a-9z - 13a-9z3 + 15a-9z5 + a-9z7 - 4a-9z9 + 2a-8z2 - 17a-8z4 + 28a-8z6 - 11a-8z8 - a-8z10 + 5a-7z - 19a-7z3 + 16a-7z5 + 4a-7z7 - 7a-7z9 + a-6z2 - 12a-6z4 + 20a-6z6 - 10a-6z8 - a-6z10 - a-5z-1 + 8a-5z - 20a-5z3 + 20a-5z5 - 6a-5z7 - 3a-5z9 + a-4 - 2a-4z2 + a-4z4 + 4a-4z6 - 5a-4z8 - a-3z-1 + 4a-3z - 6a-3z3 + 8a-3z5 - 5a-3z7 - 2a-2z2 + 5a-2z4 - 3a-2z6 - a-1z + 2a-1z3 - a-1z5

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 22                       1 j = 20                     3   j = 18                   5 1   j = 16                 7 3     j = 14               9 5       j = 12             10 7         j = 10           9 10           j = 8         7 9             j = 6       4 9               j = 4     3 7                 j = 2   1 5                   j = 0   2                     j = -2 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, 266]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 266]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 4, 13, 3], X[22, 12, 9, 11], X[2, 9, 3, 10], > X[4, 22, 5, 21], X[14, 6, 15, 5], X[16, 20, 17, 19], X[18, 8, 19, 7], > X[6, 18, 7, 17], X[20, 16, 21, 15], X[8, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, -5, 6, -9, 8, -11}, > {4, -1, 3, -2, 11, -6, 10, -7, 9, -8, 7, -10, 5, -3}]
In[6]:=	Jones[L][q]
Out[6]=	1 3/2 5/2 7/2 9/2 11/2 -(-------) + 3 Sqrt[q] - 7 q + 11 q - 16 q + 18 q - 19 q + Sqrt[q] 13/2 15/2 17/2 19/2 21/2 > 16 q - 12 q + 8 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-2 4 6 8 10 12 14 16 18 22 -1 + q + 3 q - 3 q + 3 q + 2 q - q + 4 q - q + 3 q - 2 q + 24 26 30 32 > 3 q - 3 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 266]][a, z]
Out[8]=	3 3 3 5 5 5 1 1 z 2 z 2 z z z 3 z z z 2 z z -(----) + ---- + -- - --- + --- + - + -- - ---- + -- - -- - ---- - -- 5 3 7 5 3 a 9 5 a 7 5 3 a z a z a a a a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 266]][a, z]
Out[9]=	2 2 2 2 -4 1 1 z 3 z 5 z 8 z 4 z z z 2 z z 2 z a - ---- - ---- + --- + --- + --- + --- + --- - - - --- + ---- + -- - ---- - 5 3 11 9 7 5 3 a 12 8 6 4 a z a z a a a a a a a a a 2 3 3 3 3 3 3 4 4 4 2 z 6 z 13 z 19 z 20 z 6 z 2 z 2 z 7 z 17 z > ---- - ---- - ----- - ----- - ----- - ---- + ---- + ---- - ---- - ----- - 2 11 9 7 5 3 a 12 10 8 a a a a a a a a a 4 4 4 5 5 5 5 5 5 6 12 z z 5 z 10 z 15 z 16 z 20 z 8 z z z > ----- + -- + ---- + ----- + ----- + ----- + ----- + ---- - -- - --- + 6 4 2 11 9 7 5 3 a 12 a a a a a a a a a 6 6 6 6 6 7 7 7 7 7 14 z 28 z 20 z 4 z 3 z 4 z z 4 z 6 z 5 z > ----- + ----- + ----- + ---- - ---- - ---- + -- + ---- - ---- - ---- - 10 8 6 4 2 11 9 7 5 3 a a a a a a a a a a 8 8 8 8 9 9 9 10 10 6 z 11 z 10 z 5 z 4 z 7 z 3 z z z > ---- - ----- - ----- - ---- - ---- - ---- - ---- - --- - --- 10 8 6 4 9 7 5 8 6 a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 2 q 4 6 6 2 8 2 8 3 5 q + 3 q + ----- + - + -- + 7 q t + 4 q t + 9 q t + 7 q t + 9 q t + 2 2 t t q t 10 3 10 4 12 4 12 5 14 5 14 6 > 9 q t + 10 q t + 10 q t + 7 q t + 9 q t + 5 q t + 16 6 16 7 18 7 18 8 20 8 22 9 > 7 q t + 3 q t + 5 q t + q t + 3 q t + q t

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