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L11n137 L11n139

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DrawMorseLink

PD Presentation:

X8192 X11,19,12,18 X3,10,4,11 X17,3,18,2 X5,13,6,12 X6718 X9,16,10,17 X15,20,16,21 X13,22,14,7 X21,14,22,15 X19,4,20,5

Gauss Code:

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

Jones Polynomial:

- q-17/2 + 2q-15/2 - 2q-13/2 + 3q-11/2 - 3q-9/2 + 2q-7/2 - 3q-5/2 + q-3/2 - q-1/2

A2 (sl(3)) Invariant:

q-26 - q-20 - 2q-18 + 2q-12 + 2q-10 + 2q-8 + 2q-6 + q-4 + q-2 + 1

HOMFLY-PT Polynomial:

- az-1 - az + a3z-1 + 2a3z + a3z3 - 4a5z - 4a5z3 - a5z5 + 2a7z + a7z3

Kauffman Polynomial:

az-1 - az - a2 + a3z-1 - 4a3z + 4a3z3 - a3z5 + 4a4z2 - 7a4z4 + 5a4z6 - a4z8 - 5a5z + 9a5z3 - 8a5z5 + 5a5z7 - a5z9 + 12a6z2 - 24a6z4 + 16a6z6 - 3a6z8 - a7z - a7z3 - 2a7z5 + 4a7z7 - a7z9 + 8a8z2 - 17a8z4 + 11a8z6 - 2a8z8 + a9z - 6a9z3 + 5a9z5 - a9z7

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = 0                 1 j = -2               1 1 j = -4             2     j = -6           1 2     j = -8         2 1       j = -10       1 1         j = -12     1 2           j = -14   1 1             j = -16   1               j = -18 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, 138]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 138]]
Out[4]=	PD[X[8, 1, 9, 2], X[11, 19, 12, 18], X[3, 10, 4, 11], X[17, 3, 18, 2], > X[5, 13, 6, 12], X[6, 7, 1, 8], X[9, 16, 10, 17], X[15, 20, 16, 21], > X[13, 22, 14, 7], X[21, 14, 22, 15], X[19, 4, 20, 5]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 4, -3, 11, -5, -6}, > {6, -1, -7, 3, -2, 5, -9, 10, -8, 7, -4, 2, -11, 8, -10, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 2 2 3 3 2 3 -(3/2) 1 -q + ----- - ----- + ----- - ---- + ---- - ---- + q - ------- 15/2 13/2 11/2 9/2 7/2 5/2 Sqrt[q] q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 -20 2 2 2 2 2 -4 -2 1 + q - q - --- + --- + --- + -- + -- + q + q 18 12 10 8 6 q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 138]][a, z]
Out[8]=	3 a a 3 5 7 3 3 5 3 7 3 5 5 -(-) + -- - a z + 2 a z - 4 a z + 2 a z + a z - 4 a z + a z - a z z z
In[9]:=	Kauffman[Link[11, NonAlternating, 138]][a, z]
Out[9]=	3 2 a a 3 5 7 9 4 2 6 2 -a + - + -- - a z - 4 a z - 5 a z - a z + a z + 4 a z + 12 a z + z z 8 2 3 3 5 3 7 3 9 3 4 4 6 4 > 8 a z + 4 a z + 9 a z - a z - 6 a z - 7 a z - 24 a z - 8 4 3 5 5 5 7 5 9 5 4 6 6 6 > 17 a z - a z - 8 a z - 2 a z + 5 a z + 5 a z + 16 a z + 8 6 5 7 7 7 9 7 4 8 6 8 8 8 5 9 > 11 a z + 5 a z + 4 a z - a z - a z - 3 a z - 2 a z - a z - 7 9 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	-2 1 1 1 1 1 2 1 1 + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 18 8 16 7 14 7 14 6 12 6 12 5 10 5 q t q t q t q t q t q t q t 1 2 1 1 2 2 1 > ------ + ----- + ----- + ----- + ----- + ----- + ---- 10 4 8 4 8 3 6 3 6 2 4 2 2 q t q t q t q t q t q t q t

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L11n137 L11n139