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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-1/2 + 2q1/2 - 6q3/2 + 10q5/2 - 15q7/2 + 17q9/2 - 18q11/2 + 16q13/2 - 12q15/2 + 8q17/2 - 4q19/2 + q21/2

A2 (sl(3)) Invariant:

q-2 + q2 + 4q4 - 2q6 + 3q8 + 2q10 - 2q12 + 3q14 - 2q16 + 2q18 - 2q22 + 3q24 - 3q26 + 2q30 - q32

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- a-12z2 + 2a-12z4 - a-12z6 + a-11z - 6a-11z3 + 10a-11z5 - 4a-11z7 + a-10z2 - 7a-10z4 + 14a-10z6 - 6a-10z8 + 3a-9z - 13a-9z3 + 16a-9z5 + a-9z7 - 4a-9z9 + 5a-8z2 - 23a-8z4 + 31a-8z6 - 11a-8z8 - a-8z10 + 5a-7z - 16a-7z3 + 10a-7z5 + 7a-7z7 - 7a-7z9 + 7a-6z2 - 23a-6z4 + 23a-6z6 - 9a-6z8 - a-6z10 + 3a-5z - 9a-5z3 + 7a-5z5 - a-5z7 - 3a-5z9 + 4a-4z2 - 6a-4z4 + 5a-4z6 - 4a-4z8 + a-3z-1 - 3a-3z + 3a-3z3 + 2a-3z5 - 3a-3z7 - a-2 + 3a-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                     3   j = 18                   5 1   j = 16                 7 3     j = 14               9 5       j = 12             9 7         j = 10           8 9           j = 8         7 9             j = 6       4 9               j = 4     2 6                 j = 2   1 5                   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, 242]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 242]]
Out[4]=	PD[X[8, 1, 9, 2], X[12, 4, 13, 3], X[22, 10, 7, 9], X[14, 12, 15, 11], > X[10, 16, 11, 15], X[18, 6, 19, 5], X[20, 18, 21, 17], X[16, 22, 17, 21], > X[2, 7, 3, 8], X[4, 14, 5, 13], X[6, 20, 1, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10, 6, -11}, > {9, -1, 3, -5, 4, -2, 10, -4, 5, -8, 7, -6, 11, -7, 8, -3}]
In[6]:=	Jones[L][q]
Out[6]=	1 3/2 5/2 7/2 9/2 11/2 -(-------) + 2 Sqrt[q] - 6 q + 10 q - 15 q + 17 q - 18 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 2 4 6 8 10 12 14 16 18 22 q + q + 4 q - 2 q + 3 q + 2 q - 2 q + 3 q - 2 q + 2 q - 2 q + 24 26 30 32 > 3 q - 3 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 242]][a, z]
Out[8]=	3 3 3 3 5 5 5 1 1 z z z 2 z z 3 z z z z 2 z z -(----) + --- + -- - -- - -- + --- + -- - ---- - -- + -- - -- - ---- - -- 3 a z 7 5 3 a 9 5 3 a 7 5 3 a z a a a a a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 242]][a, z]
Out[9]=	2 2 2 -2 1 1 z 3 z 5 z 3 z 3 z 3 z z z 5 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 7 z 4 z 6 z 13 z 16 z 9 z 3 z 3 z 2 z 7 z > ---- + ---- - ---- - ----- - ----- - ---- + ---- + ---- + ---- - ---- - 6 4 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 5 23 z 23 z 6 z 3 z 10 z 16 z 10 z 7 z 2 z z > ----- - ----- - ---- + ---- + ----- + ----- + ----- + ---- + ---- - -- - 8 6 4 2 11 9 7 5 3 a a a a a a a a a a 6 6 6 6 6 6 7 7 7 7 7 z 14 z 31 z 23 z 5 z 2 z 4 z z 7 z z 3 z > --- + ----- + ----- + ----- + ---- - ---- - ---- + -- + ---- - -- - ---- - 12 10 8 6 4 2 11 9 7 5 3 a a a a a a a a a a a 8 8 8 8 9 9 9 10 10 6 z 11 z 9 z 4 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 1 q 4 6 6 2 8 2 8 3 5 q + 2 q + ----- + - + -- + 6 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 > 8 q t + 9 q t + 9 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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