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L11n132 L11n134

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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 X13,20,14,21 X15,22,16,7 X19,4,20,5 X21,14,22,15

Gauss Code:

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

Jones Polynomial:

q-25/2 - q-23/2 + q-21/2 - q-19/2 - q-13/2 - q-9/2

A2 (sl(3)) Invariant:

- q-44 - q-40 - q-38 - q-36 - q-34 + q-32 + q-30 + 2q-28 + 2q-26 + 2q-24 + 2q-22 + 2q-20 + q-18 + q-16

HOMFLY-PT Polynomial:

- 3a9z-1 - 18a9z - 36a9z3 - 28a9z5 - 9a9z7 - a9z9 + 5a11z-1 + 18a11z + 20a11z3 + 8a11z5 + a11z7 - 2a13z-1 - 3a13z - a13z3

Kauffman Polynomial:

3a9z-1 - 18a9z + 36a9z3 - 28a9z5 + 9a9z7 - a9z9 - 5a10 + 18a10z2 - 20a10z4 + 8a10z6 - a10z8 + 5a11z-1 - 23a11z + 38a11z3 - 28a11z5 + 9a11z7 - a11z9 - 5a12 + 18a12z2 - 20a12z4 + 8a12z6 - a12z8 + 2a13z-1 - 4a13z + a13z3 - a14z2 + a15z - a15z3 + a16 - a16z2

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -8                   1 j = -10                   1 j = -12               1     j = -14           1         j = -16           1 1       j = -18       2 1           j = -20       1 1           j = -22   1 1               j = -24                     j = -26 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, 133]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 133]]
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[13, 20, 14, 21], > X[15, 22, 16, 7], X[19, 4, 20, 5], X[21, 14, 22, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, -3, 10, 5, -6}, > {6, -1, -7, 3, 2, -5, -8, 11, -9, 7, 4, -2, -10, 8, -11, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-(25/2) -(23/2) -(21/2) -(19/2) -(13/2) -(9/2) q - q + q - q - q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-44 -40 -38 -36 -34 -32 -30 2 2 2 2 2 -q - q - q - q - q + q + q + --- + --- + --- + --- + --- + 28 26 24 22 20 q q q q q -18 -16 > q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 133]][a, z]
Out[8]=	9 11 13 -3 a 5 a 2 a 9 11 13 9 3 11 3 ----- + ----- - ----- - 18 a z + 18 a z - 3 a z - 36 a z + 20 a z - z z z 13 3 9 5 11 5 9 7 11 7 9 9 > a z - 28 a z + 8 a z - 9 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 133]][a, z]
Out[9]=	9 11 13 10 12 16 3 a 5 a 2 a 9 11 13 -5 a - 5 a + a + ---- + ----- + ----- - 18 a z - 23 a z - 4 a z + z z z 15 10 2 12 2 14 2 16 2 9 3 11 3 > a z + 18 a z + 18 a z - a z - a z + 36 a z + 38 a z + 13 3 15 3 10 4 12 4 9 5 11 5 10 6 > a z - a z - 20 a z - 20 a z - 28 a z - 28 a z + 8 a z + 12 6 9 7 11 7 10 8 12 8 9 9 11 9 > 8 a z + 9 a z + 9 a z - a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-10 -8 1 1 1 1 2 1 1 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 26 9 22 8 22 7 20 6 18 6 20 5 18 5 q t q t q t q t q t q t q t 1 1 1 1 > ------ + ------ + ------ + ------ 16 4 14 4 16 3 12 2 q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n133

L11n132 L11n134