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DrawMorseLink

PD Presentation:

X6172 X3,10,4,11 X15,5,16,22 X7,17,8,16 X17,20,18,21 X9,14,10,15 X19,13,20,12 X13,19,14,18 X21,9,22,8 X2536 X11,4,12,1

Gauss Code:

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

Jones Polynomial:

- 3q-9/2 + 7q-7/2 - 11q-5/2 + 13q-3/2 - 15q-1/2 + 13q1/2 - 11q3/2 + 7q5/2 - 3q7/2 + q9/2

A2 (sl(3)) Invariant:

q-18 + 3q-14 - q-12 + q-10 + 2q-8 - 3q-6 + 4q-4 - q-2 + 4 + 2q2 - q4 + 2q6 - 3q8 - q14

HOMFLY-PT Polynomial:

a-3z-1 + 2a-3z + a-3z3 - 3a-1z-1 - 6a-1z - 6a-1z3 - 2a-1z5 + 3az-1 + 5az + 6az3 + 4az5 + az7 - 2a3z-1 - 4a3z - 3a3z3 - a3z5 + a5z-1 + a5z

Kauffman Polynomial:

a-4 - 3a-4z2 + 3a-4z4 - a-4z6 - a-3z-1 + 3a-3z - 7a-3z3 + 8a-3z5 - 3a-3z7 + 2a-2 - 6a-2z2 + 2a-2z4 + 7a-2z6 - 4a-2z8 - 3a-1z-1 + 12a-1z - 25a-1z3 + 27a-1z5 - 6a-1z7 - 2a-1z9 + 3z2 - 11z4 + 23z6 - 11z8 - 3az-1 + 15az - 33az3 + 35az5 - 11az7 - 2az9 - 2a2 + 9a2z2 - 13a2z4 + 12a2z6 - 7a2z8 - 2a3z-1 + 11a3z - 21a3z3 + 16a3z5 - 8a3z7 + 3a4z2 - 3a4z4 - 3a4z6 - a5z-1 + 5a5z - 6a5z3

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               5 1   j = 4             6 2     j = 2           7 5       j = 0         8 6         j = -2       6 8           j = -4     5 7             j = -6   2 6               j = -8 1 5                 j = -10 3

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, 76]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 76]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 10, 4, 11], X[15, 5, 16, 22], X[7, 17, 8, 16], > X[17, 20, 18, 21], X[9, 14, 10, 15], X[19, 13, 20, 12], X[13, 19, 14, 18], > X[21, 9, 22, 8], X[2, 5, 3, 6], X[11, 4, 12, 1]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -2, 11}, {10, -1, -4, 9, -6, 2, -11, 7, -8, 6, -3, 4, -5, 8, > -7, 5, -9, 3}]
In[6]:=	Jones[L][q]
Out[6]=	-3 7 11 13 15 3/2 5/2 7/2 ---- + ---- - ---- + ---- - ------- + 13 Sqrt[q] - 11 q + 7 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 3 -12 -10 2 3 4 -2 2 4 6 8 4 + q + --- - q + q + -- - -- + -- - q + 2 q - q + 2 q - 3 q - 14 8 6 4 q q q q 14 > q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 76]][a, z]
Out[8]=	3 5 3 3 1 3 3 a 2 a a 2 z 6 z 3 5 z 6 z ---- - --- + --- - ---- + -- + --- - --- + 5 a z - 4 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 > 6 a z - 3 a z - ---- + 4 a z - a z + a z a
In[9]:=	Kauffman[Link[11, NonAlternating, 76]][a, z]
Out[9]=	3 5 -4 2 2 1 3 3 a 2 a a 3 z 12 z a + -- - 2 a - ---- - --- - --- - ---- - -- + --- + ---- + 15 a z + 2 3 a z z z z 3 a a a z a 2 2 3 3 3 5 2 3 z 6 z 2 2 4 2 7 z 25 z > 11 a z + 5 a z + 3 z - ---- - ---- + 9 a z + 3 a z - ---- - ----- - 4 2 3 a a a a 4 4 3 3 3 5 3 4 3 z 2 z 2 4 4 4 > 33 a z - 21 a z - 6 a z - 11 z + ---- + ---- - 13 a z - 3 a z + 4 2 a a 5 5 6 6 8 z 27 z 5 3 5 6 z 7 z 2 6 > ---- + ----- + 35 a z + 16 a z + 23 z - -- + ---- + 12 a z - 3 a 4 2 a a a 7 7 8 9 4 6 3 z 6 z 7 3 7 8 4 z 2 8 2 z > 3 a z - ---- - ---- - 11 a z - 8 a z - 11 z - ---- - 7 a z - ---- - 3 a 2 a a a 9 > 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	8 3 1 5 2 6 5 7 6 8 + -- + ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + 6 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 > 7 q t + 5 q t + 6 q t + 2 q t + 5 q t + q t + 2 q t + q t

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