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DrawMorseLink

PD Presentation:

X6172 X16,9,17,10 X4,21,1,22 X11,14,12,15 X3,10,4,11 X5,13,6,12 X13,5,14,22 X15,2,16,3 X20,18,21,17 X18,8,19,7 X8,20,9,19

Gauss Code:

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

Jones Polynomial:

- 2q-9/2 + 4q-7/2 - 7q-5/2 + 9q-3/2 - 11q-1/2 + 9q1/2 - 8q3/2 + 6q5/2 - 3q7/2 + q9/2

A2 (sl(3)) Invariant:

q-18 + 2q-14 + q-10 + 2q-8 - 2q-6 + 3q-4 + 3 + 2q2 - q4 + q6 - 2q8 - q14

HOMFLY-PT Polynomial:

a-3z-1 + 2a-3z + a-3z3 - 3a-1z-1 - 7a-1z - 7a-1z3 - 2a-1z5 + 3az-1 + 7az + 9az3 + 5az5 + az7 - 2a3z-1 - 5a3z - 4a3z3 - a3z5 + a5z-1 + a5z

Kauffman Polynomial:

a-4 - 3a-4z2 + 3a-4z4 - a-4z6 - a-3z-1 + 3a-3z - 7a-3z3 + 9a-3z5 - 3a-3z7 + 2a-2 - 8a-2z2 + 6a-2z4 + 5a-2z6 - 3a-2z8 - 3a-1z-1 + 13a-1z - 27a-1z3 + 26a-1z5 - 6a-1z7 - a-1z9 - 5z4 + 13z6 - 6z8 - 3az-1 + 16az - 32az3 + 24az5 - 6az7 - az9 - 2a2 + 7a2z2 - 10a2z4 + 6a2z6 - 3a2z8 - 2a3z-1 + 10a3z - 15a3z3 + 7a3z5 - 3a3z7 + 2a4z2 - 2a4z4 - a4z6 - a5z-1 + 4a5z - 3a5z3

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 10                   1 j = 8                 2   j = 6               4 1   j = 4             4 2     j = 2           5 4       j = 0         6 4         j = -2       4 6           j = -4     3 5             j = -6   1 4               j = -8 1 3                 j = -10 2

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, NonAlternating, 79]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 79]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 9, 17, 10], X[4, 21, 1, 22], X[11, 14, 12, 15], > X[3, 10, 4, 11], X[5, 13, 6, 12], X[13, 5, 14, 22], X[15, 2, 16, 3], > X[20, 18, 21, 17], X[18, 8, 19, 7], X[8, 20, 9, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 8, -5, -3}, {-6, -1, 10, -11, 2, 5, -4, 6, -7, 4, -8, -2, 9, -10, > 11, -9, 3, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-2 4 7 9 11 3/2 5/2 7/2 ---- + ---- - ---- + ---- - ------- + 9 Sqrt[q] - 8 q + 6 q - 3 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 9/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 2 -10 2 2 3 2 4 6 8 14 3 + q + --- + q + -- - -- + -- + 2 q - q + q - 2 q - q 14 8 6 4 q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 79]][a, z]
Out[8]=	3 5 3 3 1 3 3 a 2 a a 2 z 7 z 3 5 z 7 z ---- - --- + --- - ---- + -- + --- - --- + 7 a z - 5 a z + a z + -- - ---- + 3 a z z z z 3 a 3 a a z a a 5 3 3 3 2 z 5 3 5 7 > 9 a z - 4 a z - ---- + 5 a z - a z + a z a
In[9]:=	Kauffman[Link[11, NonAlternating, 79]][a, z]
Out[9]=	3 5 -4 2 2 1 3 3 a 2 a a 3 z 13 z a + -- - 2 a - ---- - --- - --- - ---- - -- + --- + ---- + 16 a z + 2 3 a z z z z 3 a a a z a 2 2 3 3 3 5 3 z 8 z 2 2 4 2 7 z 27 z > 10 a z + 4 a z - ---- - ---- + 7 a z + 2 a z - ---- - ----- - 4 2 3 a a a a 4 4 3 3 3 5 3 4 3 z 6 z 2 4 4 4 > 32 a z - 15 a z - 3 a z - 5 z + ---- + ---- - 10 a z - 2 a z + 4 2 a a 5 5 6 6 9 z 26 z 5 3 5 6 z 5 z 2 6 4 6 > ---- + ----- + 24 a z + 7 a z + 13 z - -- + ---- + 6 a z - a z - 3 a 4 2 a a a 7 7 8 9 3 z 6 z 7 3 7 8 3 z 2 8 z 9 > ---- - ---- - 6 a z - 3 a z - 6 z - ---- - 3 a z - -- - a z 3 a 2 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	6 2 1 3 1 4 3 5 4 6 + -- + ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + 4 t + 2 10 4 8 4 8 3 6 3 6 2 4 2 4 2 q q t q t q t q t q t q t q t q t 2 2 2 4 2 4 3 6 3 6 4 8 4 10 5 > 5 q t + 4 q t + 4 q t + 2 q t + 4 q t + q t + 2 q t + q t

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