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DrawMorseLink

PD Presentation:

X6172 X20,7,21,8 X4,21,1,22 X9,14,10,15 X8493 X12,5,13,6 X22,13,5,14 X15,18,16,19 X11,17,12,16 X17,11,18,10 X2,20,3,19

Gauss Code:

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

Jones Polynomial:

q-15/2 - 4q-13/2 + 7q-11/2 - 10q-9/2 + 12q-7/2 - 12q-5/2 + 11q-3/2 - 9q-1/2 + 4q1/2 - 2q3/2

A2 (sl(3)) Invariant:

- q-24 + 3q-20 - q-18 + 2q-16 + q-14 - 3q-12 + q-10 - 3q-8 + q-6 + 5 + 2q4 + 2q6

HOMFLY-PT Polynomial:

- 2a-1z-1 - 2a-1z + 4az-1 + 7az + 4az3 - 3a3z-1 - 8a3z - 6a3z3 - 2a3z5 + a5z-1 + 4a5z + 3a5z3 - a7z

Kauffman Polynomial:

- 2a-1z-1 + 5a-1z - 3a-1z3 + 2 - 2z4 - z6 - 4az-1 + 15az - 21az3 + 10az5 - 4az7 + 3a2 - a2z2 - 13a2z4 + 12a2z6 - 5a2z8 - 3a3z-1 + 15a3z - 33a3z3 + 26a3z5 - 5a3z7 - 2a3z9 + 3a4 - 17a4z4 + 26a4z6 - 10a4z8 - a5z-1 + 6a5z - 22a5z3 + 27a5z5 - 5a5z7 - 2a5z9 + a6 - 4a6z4 + 12a6z6 - 5a6z8 + a7z - 7a7z3 + 11a7z5 - 4a7z7 - a8z2 + 2a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 4                   2 j = 2                 2   j = 0               7 2   j = -2             6 4     j = -4           6 5       j = -6         6 6         j = -8       4 6           j = -10     3 6             j = -12   1 4               j = -14   3                 j = -16 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, NonAlternating, 35]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 35]]
Out[4]=	PD[X[6, 1, 7, 2], X[20, 7, 21, 8], X[4, 21, 1, 22], X[9, 14, 10, 15], > X[8, 4, 9, 3], X[12, 5, 13, 6], X[22, 13, 5, 14], X[15, 18, 16, 19], > X[11, 17, 12, 16], X[17, 11, 18, 10], X[2, 20, 3, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {6, -1, 2, -5, -4, 10, -9, -6, 7, 4, -8, 9, -10, 8, > 11, -2, 3, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 4 7 10 12 12 11 9 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 4 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 > 2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 3 -18 2 -14 3 -10 3 -6 4 6 5 - q + --- - q + --- + q - --- + q - -- + q + 2 q + 2 q 20 16 12 8 q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 35]][a, z]
Out[8]=	3 5 -2 4 a 3 a a 2 z 3 5 7 3 --- + --- - ---- + -- - --- + 7 a z - 8 a z + 4 a z - a z + 4 a z - a z z z z a 3 3 5 3 3 5 > 6 a z + 3 a z - 2 a z
In[9]:=	Kauffman[Link[11, NonAlternating, 35]][a, z]
Out[9]=	3 5 2 4 6 2 4 a 3 a a 5 z 3 2 + 3 a + 3 a + a - --- - --- - ---- - -- + --- + 15 a z + 15 a z + a z z z z a 3 5 7 2 2 8 2 3 z 3 3 3 5 3 > 6 a z + a z - a z - a z - ---- - 21 a z - 33 a z - 22 a z - a 7 3 4 2 4 4 4 6 4 8 4 5 > 7 a z - 2 z - 13 a z - 17 a z - 4 a z + 2 a z + 10 a z + 3 5 5 5 7 5 6 2 6 4 6 6 6 > 26 a z + 27 a z + 11 a z - z + 12 a z + 26 a z + 12 a z - 8 6 7 3 7 5 7 7 7 2 8 4 8 > a z - 4 a z - 5 a z - 5 a z - 4 a z - 5 a z - 10 a z - 6 8 3 9 5 9 > 5 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 1 3 1 4 3 6 4 6 7 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 6 6 6 5 6 2 4 2 > ----- + ----- + ----- + ---- + ---- + 2 t + 2 q t + 2 q t 6 3 6 2 4 2 4 2 q t q t q t q t q t

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