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L11n140 L11n142

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DrawMorseLink

PD Presentation:

X8192 X18,11,19,12 X3,10,4,11 X2,17,3,18 X12,5,13,6 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-21/2 - q-19/2 + q-17/2 - 2q-15/2 + q-13/2 - 2q-11/2 + q-9/2 - q-7/2

A2 (sl(3)) Invariant:

- q-42 - q-36 - q-32 + 2q-26 + 3q-24 + 2q-22 + 2q-20 + q-18 + q-16 + q-12

HOMFLY-PT Polynomial:

- a7z-1 - 5a7z - 10a7z3 - 6a7z5 - a7z7 - 5a9z - 10a9z3 - 6a9z5 - a9z7 + 2a11z-1 + 8a11z + 6a11z3 + a11z5 - a13z-1 - a13z

Kauffman Polynomial:

- a7z-1 + 5a7z - 10a7z3 + 6a7z5 - a7z7 + a8 - a8z2 - 5a8z4 + 5a8z6 - a8z8 - 3a9z + 9a9z3 - 11a9z5 + 6a9z7 - a9z9 - 3a10 + 14a10z2 - 21a10z4 + 12a10z6 - 2a10z8 + 2a11z-1 - 10a11z + 20a11z3 - 17a11z5 + 7a11z7 - a11z9 - 5a12 + 16a12z2 - 16a12z4 + 7a12z6 - a12z8 + a13z-1 - 2a13z + a13z3 - 2a14 + a14z2

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -6               1 j = -8             1 1 j = -10           1     j = -12       1 1 1     j = -14       2 1       j = -16   1 1 1         j = -18   2 2           j = -20                 j = -22 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, 141]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 141]]
Out[4]=	PD[X[8, 1, 9, 2], X[18, 11, 19, 12], X[3, 10, 4, 11], X[2, 17, 3, 18], > X[12, 5, 13, 6], 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]=	-(21/2) -(19/2) -(17/2) 2 -(13/2) 2 -(9/2) -(7/2) q - q + q - ----- + q - ----- + q - q 15/2 11/2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 -36 -32 2 3 2 2 -18 -16 -12 -q - q - q + --- + --- + --- + --- + q + q + q 26 24 22 20 q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 141]][a, z]
Out[8]=	7 11 13 a 2 a a 7 9 11 13 7 3 9 3 -(--) + ----- - --- - 5 a z - 5 a z + 8 a z - a z - 10 a z - 10 a z + z z z 11 3 7 5 9 5 11 5 7 7 9 7 > 6 a z - 6 a z - 6 a z + a z - a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 141]][a, z]
Out[9]=	7 11 13 8 10 12 14 a 2 a a 7 9 11 a - 3 a - 5 a - 2 a - -- + ----- + --- + 5 a z - 3 a z - 10 a z - z z z 13 8 2 10 2 12 2 14 2 7 3 9 3 > 2 a z - a z + 14 a z + 16 a z + a z - 10 a z + 9 a z + 11 3 13 3 8 4 10 4 12 4 7 5 9 5 > 20 a z + a z - 5 a z - 21 a z - 16 a z + 6 a z - 11 a z - 11 5 8 6 10 6 12 6 7 7 9 7 11 7 > 17 a z + 5 a z + 12 a z + 7 a z - a z + 6 a z + 7 a z - 8 8 10 8 12 8 9 9 11 9 > a z - 2 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-8 -6 1 2 1 2 1 1 2 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 22 7 18 6 16 6 18 5 16 5 16 4 14 4 q t q t q t q t q t q t q t 1 1 1 1 1 1 > ------ + ------ + ------ + ------ + ------ + ---- 12 4 14 3 12 3 12 2 10 2 8 q t q t q t q t q t q t

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