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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- 2q-15/2 + 5q-13/2 - 9q-11/2 + 12q-9/2 - 13q-7/2 + 12q-5/2 - 11q-3/2 + 7q-1/2 - 4q1/2 + q3/2

A2 (sl(3)) Invariant:

q-28 + q-26 + q-24 + q-22 - 3q-20 + q-18 - 2q-16 + 3q-12 - q-10 + 5q-8 - q-6 + 2q-4 + q-2 - 1 + 2q2 - q4

HOMFLY-PT Polynomial:

2az3 + az5 - 2a3z-1 - 6a3z - 6a3z3 - 4a3z5 - a3z7 + 4a5z-1 + 8a5z + 7a5z3 + 2a5z5 - 3a7z-1 - 4a7z - a7z3 + a9z-1

Kauffman Polynomial:

- z2 + 2z4 - z6 + 2az - 8az3 + 11az5 - 4az7 - a2 - 2a2z4 + 11a2z6 - 5a2z8 - 2a3z-1 + 15a3z - 35a3z3 + 35a3z5 - 7a3z7 - 2a3z9 - 2a4 + 9a4z2 - 23a4z4 + 29a4z6 - 11a4z8 - 4a5z-1 + 25a5z - 49a5z3 + 37a5z5 - 8a5z7 - 2a5z9 - 3a6 + 12a6z2 - 23a6z4 + 16a6z6 - 6a6z8 - 3a7z-1 + 16a7z - 25a7z3 + 13a7z5 - 5a7z7 - a8 + 4a8z2 - 4a8z4 - a8z6 - a9z-1 + 4a9z - 3a9z3

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 4                   1 j = 2                 3   j = 0               4 1   j = -2             7 3     j = -4           6 5       j = -6         7 6         j = -8       5 6           j = -10     4 7             j = -12   2 6               j = -14   3                 j = -16 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, 66]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 66]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 10, 4, 11], X[7, 14, 8, 15], X[15, 22, 16, 5], > X[9, 17, 10, 16], X[21, 9, 22, 8], X[17, 21, 18, 20], X[13, 18, 14, 19], > X[19, 12, 20, 13], X[2, 5, 3, 6], X[11, 4, 12, 1]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -2, 11}, {10, -1, -3, 6, -5, 2, -11, 9, -8, 3, -4, 5, -7, 8, > -9, 7, -6, 4}]
In[6]:=	Jones[L][q]
Out[6]=	-2 5 9 12 13 12 11 7 3/2 ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 4 Sqrt[q] + q 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 -26 -24 -22 3 -18 2 3 -10 5 -6 -1 + q + q + q + q - --- + q - --- + --- - q + -- - q + 20 16 12 8 q q q q 2 -2 2 4 > -- + q + 2 q - q 4 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 66]][a, z]
Out[8]=	3 5 7 9 -2 a 4 a 3 a a 3 5 7 3 3 3 ----- + ---- - ---- + -- - 6 a z + 8 a z - 4 a z + 2 a z - 6 a z + z z z z 5 3 7 3 5 3 5 5 5 3 7 > 7 a z - a z + a z - 4 a z + 2 a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 66]][a, z]
Out[9]=	3 5 7 9 2 4 6 8 2 a 4 a 3 a a 3 5 -a - 2 a - 3 a - a - ---- - ---- - ---- - -- + 2 a z + 15 a z + 25 a z + z z z z 7 9 2 4 2 6 2 8 2 3 3 3 > 16 a z + 4 a z - z + 9 a z + 12 a z + 4 a z - 8 a z - 35 a z - 5 3 7 3 9 3 4 2 4 4 4 6 4 > 49 a z - 25 a z - 3 a z + 2 z - 2 a z - 23 a z - 23 a z - 8 4 5 3 5 5 5 7 5 6 2 6 > 4 a z + 11 a z + 35 a z + 37 a z + 13 a z - z + 11 a z + 4 6 6 6 8 6 7 3 7 5 7 7 7 > 29 a z + 16 a z - a z - 4 a z - 7 a z - 8 a z - 5 a z - 2 8 4 8 6 8 3 9 5 9 > 5 a z - 11 a z - 6 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 7 2 3 2 6 4 7 5 6 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 4 2 16 6 14 5 12 5 12 4 10 4 10 3 8 3 8 2 q q q t q t q t q t q t q t q t q t 7 6 6 3 t 2 2 2 4 3 > ----- + ---- + ---- + 4 t + --- + t + 3 q t + q t 6 2 6 4 2 q t q t q t q

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