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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-1/2 + 3q1/2 - 7q3/2 + 11q5/2 - 15q7/2 + 17q9/2 - 17q11/2 + 14q13/2 - 11q15/2 + 6q17/2 - 3q19/2 + q21/2

A2 (sl(3)) Invariant:

q-2 - 1 + 3q4 - 3q6 + 2q8 - 2q12 + 3q14 - q16 + 4q18 + q20 + 4q24 - 2q26 + q30 - q32

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- 2a-12z2 + 3a-12z4 - a-12z6 + 2a-11z - 8a-11z3 + 9a-11z5 - 3a-11z7 - a-10z2 - 4a-10z4 + 9a-10z6 - 4a-10z8 + 3a-9z - 6a-9z3 + 6a-9z5 + 2a-9z7 - 3a-9z9 - 5a-8z4 + 13a-8z6 - 6a-8z8 - a-8z10 + a-7z-1 - 5a-7z + 6a-7z3 - 3a-7z5 + 7a-7z7 - 6a-7z9 - a-6 + a-6z2 - 4a-6z4 + 11a-6z6 - 7a-6z8 - a-6z10 + a-5z-1 - 3a-5z - 3a-5z3 + 9a-5z5 - 3a-5z7 - 3a-5z9 - a-4z4 + 5a-4z6 - 5a-4z8 + 2a-3z - 5a-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                     2   j = 18                   4 1   j = 16                 7 2     j = 14               8 5       j = 12             9 6         j = 10           8 8           j = 8         7 9             j = 6       4 8               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, 363]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 363]]
Out[4]=	PD[X[12, 1, 13, 2], X[14, 4, 15, 3], X[22, 14, 11, 13], X[2, 11, 3, 12], > X[4, 22, 5, 21], X[20, 10, 21, 9], X[16, 6, 17, 5], X[8, 18, 9, 17], > X[18, 8, 19, 7], X[6, 20, 7, 19], X[10, 16, 1, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, -5, 7, -10, 9, -8, 6, -11}, > {4, -1, 3, -2, 11, -7, 8, -9, 10, -6, 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 - 15 q + 17 q - 17 q + Sqrt[q] 13/2 15/2 17/2 19/2 21/2 > 14 q - 11 q + 6 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-2 4 6 8 12 14 16 18 20 24 -1 + q + 3 q - 3 q + 2 q - 2 q + 3 q - q + 4 q + q + 4 q - 26 30 32 > 2 q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 363]][a, z]
Out[8]=	3 3 3 3 5 5 5 1 1 z z 2 z z z z 3 z z z 2 z z -(----) + ---- + -- - -- + --- + - + -- - -- - ---- + -- - -- - ---- - -- 7 5 9 7 3 a 9 7 5 a 7 5 3 a z a z a a a a a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 363]][a, z]
Out[9]=	2 2 2 2 -6 1 1 2 z 3 z 5 z 3 z 2 z z 2 z z z 2 z -a + ---- + ---- + --- + --- - --- - --- + --- - - - ---- - --- + -- - ---- - 7 5 11 9 7 5 3 a 12 10 6 2 a z a z a a a a a a a a a 3 3 3 3 3 3 4 4 4 4 4 8 z 6 z 6 z 3 z 5 z 2 z 3 z 4 z 5 z 4 z z > ---- - ---- + ---- - ---- - ---- + ---- + ---- - ---- - ---- - ---- - -- + 11 9 7 5 3 a 12 10 8 6 4 a a a a a a a a a a 4 5 5 5 5 5 5 6 6 6 6 5 z 9 z 6 z 3 z 9 z 8 z z z 9 z 13 z 11 z > ---- + ---- + ---- - ---- + ---- + ---- - -- - --- + ---- + ----- + ----- + 2 11 9 7 5 3 a 12 10 8 6 a a a a a a a a a a 6 6 7 7 7 7 7 8 8 8 5 z 3 z 3 z 2 z 7 z 3 z 5 z 4 z 6 z 7 z > ---- - ---- - ---- + ---- + ---- - ---- - ---- - ---- - ---- - ---- - 4 2 11 9 7 5 3 10 8 6 a a a a a a a a a a 8 9 9 9 10 10 5 z 3 z 6 z 3 z z z > ---- - ---- - ---- - ---- - --- - --- 4 9 7 5 8 6 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 + 8 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 + 8 q t + 9 q t + 6 q t + 8 q t + 5 q t + 16 6 16 7 18 7 18 8 20 8 22 9 > 7 q t + 2 q t + 4 q t + q t + 2 q t + q t

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