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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X14,8,15,7 X22,16,5,15 X16,11,17,12 X20,18,21,17 X18,10,19,9 X10,20,11,19 X8,21,9,22 X2536 X4,14,1,13

Gauss Code:

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

Jones Polynomial:

- q-5/2 + 3q-3/2 - 7q-1/2 + 11q1/2 - 16q3/2 + 18q5/2 - 20q7/2 + 17q9/2 - 13q11/2 + 9q13/2 - 4q15/2 + q17/2

A2 (sl(3)) Invariant:

q-8 + 3q-2 - 2 + 2q2 + 2q4 - 2q6 + 5q8 - q10 + 4q12 + q14 - 2q16 + 2q18 - 4q20 + q24 - q26

HOMFLY-PT Polynomial:

a-7z-1 + a-7z + a-7z3 - 3a-5z-1 - 5a-5z - 5a-5z3 - 2a-5z5 + 3a-3z-1 + 8a-3z + 8a-3z3 + 4a-3z5 + a-3z7 - 2a-1z-1 - 6a-1z - 6a-1z3 - 2a-1z5 + az-1 + 2az + az3

Kauffman Polynomial:

- a-10z4 + a-9z3 - 4a-9z5 + a-8 - 4a-8z2 + 9a-8z4 - 9a-8z6 - a-7z-1 + 3a-7z - 9a-7z3 + 16a-7z5 - 12a-7z7 + 2a-6 - 5a-6z2 + 7a-6z4 + 8a-6z6 - 10a-6z8 - 3a-5z-1 + 12a-5z - 31a-5z3 + 39a-5z5 - 9a-5z7 - 5a-5z9 + 7a-4z2 - 31a-4z4 + 46a-4z6 - 17a-4z8 - a-4z10 - 3a-3z-1 + 18a-3z - 41a-3z3 + 28a-3z5 + 8a-3z7 - 8a-3z9 - 2a-2 + 13a-2z2 - 41a-2z4 + 40a-2z6 - 10a-2z8 - a-2z10 - 2a-1z-1 + 13a-1z - 26a-1z3 + 13a-1z5 + 4a-1z7 - 3a-1z9 + 5z2 - 13z4 + 11z6 - 3z8 - az-1 + 4az - 6az3 + 4az5 - az7

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 18                       1 j = 16                     3   j = 14                   6 1   j = 12                 7 3     j = 10               10 6       j = 8             10 7         j = 6           8 10           j = 4         8 10             j = 2       5 10               j = 0     2 6                 j = -2   1 5                   j = -4   2                     j = -6 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, 63]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 63]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[14, 8, 15, 7], X[22, 16, 5, 15], > X[16, 11, 17, 12], X[20, 18, 21, 17], X[18, 10, 19, 9], X[10, 20, 11, 19], > X[8, 21, 9, 22], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -9, 7, -8, 5, -2, 11, -3, 4, -5, 6, -7, > 8, -6, 9, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 3 7 3/2 5/2 7/2 -q + ---- - ------- + 11 Sqrt[q] - 16 q + 18 q - 20 q + 3/2 Sqrt[q] q 9/2 11/2 13/2 15/2 17/2 > 17 q - 13 q + 9 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 3 2 4 6 8 10 12 14 16 18 -2 + q + -- + 2 q + 2 q - 2 q + 5 q - q + 4 q + q - 2 q + 2 q - 2 q 20 24 26 > 4 q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 63]][a, z]
Out[8]=	3 3 1 3 3 2 a z 5 z 8 z 6 z z 5 z ---- - ---- + ---- - --- + - + -- - --- + --- - --- + 2 a z + -- - ---- + 7 5 3 a z z 7 5 3 a 7 5 a z a z a z a a a a a 3 3 5 5 5 7 8 z 6 z 3 2 z 4 z 2 z z > ---- - ---- + a z - ---- + ---- - ---- + -- 3 a 5 3 a 3 a a a a
In[9]:=	Kauffman[Link[11, Alternating, 63]][a, z]
Out[9]=	-8 2 2 1 3 3 2 a 3 z 12 z 18 z 13 z a + -- - -- - ---- - ---- - ---- - --- - - + --- + ---- + ---- + ---- + 6 2 7 5 3 a z z 7 5 3 a a a a z a z a z a a a 2 2 2 2 3 3 3 3 2 4 z 5 z 7 z 13 z z 9 z 31 z 41 z > 4 a z + 5 z - ---- - ---- + ---- + ----- + -- - ---- - ----- - ----- - 8 6 4 2 9 7 5 3 a a a a a a a a 3 4 4 4 4 4 5 5 26 z 3 4 z 9 z 7 z 31 z 41 z 4 z 16 z > ----- - 6 a z - 13 z - --- + ---- + ---- - ----- - ----- - ---- + ----- + a 10 8 6 4 2 9 7 a a a a a a a 5 5 5 6 6 6 6 39 z 28 z 13 z 5 6 9 z 8 z 46 z 40 z > ----- + ----- + ----- + 4 a z + 11 z - ---- + ---- + ----- + ----- - 5 3 a 8 6 4 2 a a a a a a 7 7 7 7 8 8 8 9 12 z 9 z 8 z 4 z 7 8 10 z 17 z 10 z 5 z > ----- - ---- + ---- + ---- - a z - 3 z - ----- - ----- - ----- - ---- - 7 5 3 a 6 4 2 5 a a a a a a a 9 9 10 10 8 z 3 z z z > ---- - ---- - --- - --- 3 a 4 2 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 2 1 2 5 6 5 q 4 10 q + 8 q + ----- + ----- + ----- + -- + ----- + - + ---- + 10 q t + 6 4 4 3 2 3 2 2 2 t t q t q t q t t q t 6 6 2 8 2 8 3 10 3 10 4 12 4 > 8 q t + 10 q t + 10 q t + 7 q t + 10 q t + 6 q t + 7 q t + 12 5 14 5 14 6 16 6 18 7 > 3 q t + 6 q t + q t + 3 q t + q t

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