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DrawMorseLink

PD Presentation:

X8192 X10,4,11,3 X22,10,7,9 X16,6,17,5 X20,14,21,13 X18,16,19,15 X14,20,15,19 X12,22,13,21 X2738 X4,12,5,11 X6,18,1,17

Gauss Code:

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

Jones Polynomial:

- q-1/2 + 2q1/2 - 5q3/2 + 7q5/2 - 11q7/2 + 12q9/2 - 12q11/2 + 11q13/2 - 8q15/2 + 5q17/2 - 3q19/2 + q21/2

A2 (sl(3)) Invariant:

q-2 + q2 + 3q4 - q6 + 3q8 + 2q10 + 2q14 - 2q16 - q20 - q22 + 3q24 - q26 + q30 - q32

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- a-12z2 + 3a-12z4 - a-12z6 + a-11z - 7a-11z3 + 10a-11z5 - 3a-11z7 + 4a-10z2 - 11a-10z4 + 13a-10z6 - 4a-10z8 + a-9z - 4a-9z3 - a-9z5 + 7a-9z7 - 3a-9z9 + 9a-8z2 - 23a-8z4 + 17a-8z6 - 3a-8z8 - a-8z10 - a-7z + 7a-7z3 - 19a-7z5 + 15a-7z7 - 5a-7z9 + 5a-6z2 - 10a-6z4 + 5a-6z6 - a-6z8 - a-6z10 - 3a-5z + 8a-5z3 - 7a-5z5 + 3a-5z7 - 2a-5z9 + 3a-4z4 - 2a-4z8 + a-3z-1 - 5a-3z + 7a-3z3 - 2a-3z7 - a-2 - a-2z2 + 4a-2z4 - 2a-2z6 + a-1z-1 - 3a-1z + 3a-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                     2   j = 18                   3 1   j = 16                 5 2     j = 14               6 3       j = 12             6 5         j = 10           6 6           j = 8         5 6             j = 6       3 7               j = 4     2 4                 j = 2   1 4                   j = 0   1                     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, 194]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 194]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[22, 10, 7, 9], X[16, 6, 17, 5], > X[20, 14, 21, 13], X[18, 16, 19, 15], X[14, 20, 15, 19], X[12, 22, 13, 21], > X[2, 7, 3, 8], X[4, 12, 5, 11], X[6, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10, 4, -11}, > {9, -1, 3, -2, 10, -8, 5, -7, 6, -4, 11, -6, 7, -5, 8, -3}]
In[6]:=	Jones[L][q]
Out[6]=	1 3/2 5/2 7/2 9/2 11/2 -(-------) + 2 Sqrt[q] - 5 q + 7 q - 11 q + 12 q - 12 q + Sqrt[q] 13/2 15/2 17/2 19/2 21/2 > 11 q - 8 q + 5 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-2 2 4 6 8 10 14 16 20 22 24 26 q + q + 3 q - q + 3 q + 2 q + 2 q - 2 q - q - q + 3 q - q + 30 32 > q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 194]][a, z]
Out[8]=	3 3 3 3 5 5 1 1 z 2 z 2 z 2 z 2 z z 2 z 2 z z z z -(----) + --- + -- - --- + --- - --- + --- + -- - ---- - ---- + -- - -- - -- - 3 a z 9 7 5 3 a 9 7 3 a 7 5 a z a a a a a a a a a 5 z > -- 3 a
In[9]:=	Kauffman[Link[11, Alternating, 194]][a, z]
Out[9]=	2 2 2 -2 1 1 z z z 3 z 5 z 3 z z 4 z 9 z -a + ---- + --- + --- + -- - -- - --- - --- - --- - --- + ---- + ---- + 3 a z 11 9 7 5 3 a 12 10 8 a z a a a a a a a a 2 2 3 3 3 3 3 3 4 4 5 z z 7 z 4 z 7 z 8 z 7 z 3 z 3 z 11 z > ---- - -- - ---- - ---- + ---- + ---- + ---- + ---- + ---- - ----- - 6 2 11 9 7 5 3 a 12 10 a a a a a a a a a 4 4 4 4 5 5 5 5 5 6 23 z 10 z 3 z 4 z 10 z z 19 z 7 z z z > ----- - ----- + ---- + ---- + ----- - -- - ----- - ---- - -- - --- + 8 6 4 2 11 9 7 5 a 12 a a a a a a a a a 6 6 6 6 7 7 7 7 7 8 13 z 17 z 5 z 2 z 3 z 7 z 15 z 3 z 2 z 4 z > ----- + ----- + ---- - ---- - ---- + ---- + ----- + ---- - ---- - ---- - 10 8 6 2 11 9 7 5 3 10 a a a a a a a a a a 8 8 8 9 9 9 10 10 3 z z 2 z 3 z 5 z 2 z z z > ---- - -- - ---- - ---- - ---- - ---- - --- - --- 8 6 4 9 7 5 8 6 a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 1 q 4 6 6 2 8 2 8 3 4 q + 2 q + ----- + - + -- + 4 q t + 3 q t + 7 q t + 5 q t + 6 q t + 2 2 t t q t 10 3 10 4 12 4 12 5 14 5 14 6 > 6 q t + 6 q t + 6 q t + 5 q t + 6 q t + 3 q t + 16 6 16 7 18 7 18 8 20 8 22 9 > 5 q t + 2 q t + 3 q t + q t + 2 q t + q t

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