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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 2q-5/2 - 6q-3/2 + 9q-1/2 - 12q1/2 + 14q3/2 - 14q5/2 + 12q7/2 - 9q9/2 + 5q11/2 - 3q13/2 + q15/2

A2 (sl(3)) Invariant:

q-12 + q-10 + 4q-6 + q-4 - q-2 + 3 - 3q2 - q6 - q8 + 2q10 - 2q12 + 3q14 + 2q16 - q18 + 2q20 - q24

HOMFLY-PT Polynomial:

a-7z - a-5z-1 - 2a-5z - 2a-5z3 + 2a-3z-1 + 3a-3z + a-3z3 + a-3z5 - a-1z-1 - a-1z + a-1z5 - az-1 - 2az - 2az3 + a3z-1 + a3z

Kauffman Polynomial:

- a-8z2 + 3a-8z4 - a-8z6 + 2a-7z - 8a-7z3 + 10a-7z5 - 3a-7z7 + 2a-6z2 - 9a-6z4 + 12a-6z6 - 4a-6z8 - a-5z-1 + 7a-5z - 13a-5z3 + 5a-5z5 + 5a-5z7 - 3a-5z9 - a-4 + 5a-4z2 - 17a-4z4 + 16a-4z6 - 4a-4z8 - a-4z10 - 2a-3z-1 + 13a-3z - 15a-3z3 - a-3z5 + 9a-3z7 - 5a-3z9 - 3a-2 + 7a-2z2 - 9a-2z4 + 7a-2z6 - 3a-2z8 - a-2z10 - a-1z-1 + 8a-1z - 11a-1z3 + 8a-1z5 - 2a-1z7 - 2a-1z9 - 2 + 5z2 - z4 + 2z6 - 3z8 + az-1 - 3az + 2az3 + 3az5 - 3az7 - a2 + 3a2z4 - 2a2z6 + a3z-1 - 3a3z + 3a3z3 - a3z5

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 = 16                       1 j = 14                     2   j = 12                   3 1   j = 10                 6 2     j = 8               6 3       j = 6             8 6         j = 4           6 6           j = 2         6 8             j = 0       5 8               j = -2     1 4                 j = -4   1 5                   j = -6   1                     j = -8 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, 96]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 96]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[20, 8, 21, 7], X[22, 15, 5, 16], > X[16, 21, 17, 22], X[18, 10, 19, 9], X[14, 12, 15, 11], X[10, 18, 11, 17], > X[8, 20, 9, 19], 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, 6, -8, 7, -2, 11, -7, 4, -5, 8, -6, > 9, -3, 5, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 2 6 9 3/2 5/2 7/2 -q + ---- - ---- + ------- - 12 Sqrt[q] + 14 q - 14 q + 12 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 13/2 15/2 > 9 q + 5 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -10 4 -4 -2 2 6 8 10 12 14 3 + q + q + -- + q - q - 3 q - q - q + 2 q - 2 q + 3 q + 6 q 16 18 20 24 > 2 q - q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 96]][a, z]
Out[8]=	3 3 3 1 2 1 a a z 2 z 3 z z 3 2 z z -(----) + ---- - --- - - + -- + -- - --- + --- - - - 2 a z + a z - ---- + -- - 5 3 a z z z 7 5 3 a 5 3 a z a z a a a a a 5 5 3 z z > 2 a z + -- + -- 3 a a
In[9]:=	Kauffman[Link[11, Alternating, 96]][a, z]
Out[9]=	3 -4 3 2 1 2 1 a a 2 z 7 z 13 z 8 z -2 - a - -- - a - ---- - ---- - --- + - + -- + --- + --- + ---- + --- - 2 5 3 a z z z 7 5 3 a a a z a z a a a 2 2 2 2 3 3 3 3 2 z 2 z 5 z 7 z 8 z 13 z 15 z > 3 a z - 3 a z + 5 z - -- + ---- + ---- + ---- - ---- - ----- - ----- - 8 6 4 2 7 5 3 a a a a a a a 3 4 4 4 4 11 z 3 3 3 4 3 z 9 z 17 z 9 z 2 4 > ----- + 2 a z + 3 a z - z + ---- - ---- - ----- - ---- + 3 a z + a 8 6 4 2 a a a a 5 5 5 5 6 6 6 10 z 5 z z 8 z 5 3 5 6 z 12 z 16 z > ----- + ---- - -- + ---- + 3 a z - a z + 2 z - -- + ----- + ----- + 7 5 3 a 8 6 4 a a a a a a 6 7 7 7 7 8 8 7 z 2 6 3 z 5 z 9 z 2 z 7 8 4 z 4 z > ---- - 2 a z - ---- + ---- + ---- - ---- - 3 a z - 3 z - ---- - ---- - 2 7 5 3 a 6 4 a a a a a a 8 9 9 9 10 10 3 z 3 z 5 z 2 z z z > ---- - ---- - ---- - ---- - --- - --- 2 5 3 a 4 2 a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 5 1 5 4 2 4 8 + 6 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 8 q t + 6 q t + 8 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t 4 2 6 2 6 3 8 3 8 4 10 4 10 5 > 6 q t + 8 q t + 6 q t + 6 q t + 3 q t + 6 q t + 2 q t + 12 5 12 6 14 6 16 7 > 3 q t + q t + 2 q t + q t

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