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L11n135 L11n137

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DrawMorseLink

PD Presentation:

X8192 X11,19,12,18 X3,10,4,11 X2,17,3,18 X12,5,13,6 X6718 X16,10,17,9 X20,14,21,13 X22,16,7,15 X19,4,20,5 X14,22,15,21

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-11/2 + q-9/2 - 2q-7/2 + 2q-5/2 - 3q-3/2 + 2q-1/2 - 3q1/2 + 2q3/2 - q5/2 + q7/2

A2 (sl(3)) Invariant:

q-16 + q-14 + 2q-12 + 2q-10 + 2q-8 + 2q-6 + q-4 + 2q-2 + 1 - q4 - 2q6 - q8 - q10

HOMFLY-PT Polynomial:

2a-1z-1 + 7a-1z + 5a-1z3 + a-1z5 - 5az-1 - 17az - 17az3 - 7az5 - az7 + 3a3z-1 + 7a3z + 5a3z3 + a3z5

Kauffman Polynomial:

a-4 - a-4z2 - a-3z3 - a-2z4 + 2a-1z-1 - 7a-1z + 6a-1z3 - 2a-1z5 - 5 + 17z2 - 15z4 + 6z6 - z8 + 5az-1 - 22az + 33az3 - 21az5 + 7az7 - az9 - 5a2 + 17a2z2 - 20a2z4 + 11a2z6 - 2a2z8 + 3a3z-1 - 11a3z + 16a3z3 - 13a3z5 + 6a3z7 - a3z9 + a4z2 - 6a4z4 + 5a4z6 - a4z8 + 4a5z - 10a5z3 + 6a5z5 - a5z7

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 8                   1 j = 6                     j = 4               2 1   j = 2             1       j = 0           2 3       j = -2         1           j = -4       1 2           j = -6     1 1             j = -8     1               j = -10 1 1                 j = -12 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, 136]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 136]]
Out[4]=	PD[X[8, 1, 9, 2], X[11, 19, 12, 18], X[3, 10, 4, 11], X[2, 17, 3, 18], > X[12, 5, 13, 6], X[6, 7, 1, 8], X[16, 10, 17, 9], X[20, 14, 21, 13], > X[22, 16, 7, 15], X[19, 4, 20, 5], X[14, 22, 15, 21]]
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]=	-(11/2) -(9/2) 2 2 3 2 3/2 -q + q - ---- + ---- - ---- + ------- - 3 Sqrt[q] + 2 q - 7/2 5/2 3/2 Sqrt[q] q q q 5/2 7/2 > q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -14 2 2 2 2 -4 2 4 6 8 10 1 + q + q + --- + --- + -- + -- + q + -- - q - 2 q - q - q 12 10 8 6 2 q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 136]][a, z]
Out[8]=	3 3 5 2 5 a 3 a 7 z 3 5 z 3 3 3 z --- - --- + ---- + --- - 17 a z + 7 a z + ---- - 17 a z + 5 a z + -- - a z z z a a a 5 3 5 7 > 7 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 136]][a, z]
Out[9]=	3 -4 2 2 5 a 3 a 7 z 3 5 2 -5 + a - 5 a + --- + --- + ---- - --- - 22 a z - 11 a z + 4 a z + 17 z - a z z z a 2 3 3 z 2 2 4 2 z 6 z 3 3 3 5 3 4 > -- + 17 a z + a z - -- + ---- + 33 a z + 16 a z - 10 a z - 15 z - 4 3 a a a 4 5 z 2 4 4 4 2 z 5 3 5 5 5 6 > -- - 20 a z - 6 a z - ---- - 21 a z - 13 a z + 6 a z + 6 z + 2 a a 2 6 4 6 7 3 7 5 7 8 2 8 4 8 > 11 a z + 5 a z + 7 a z + 6 a z - a z - z - 2 a z - a z - 9 3 9 > a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 1 1 1 1 2 3 + q + ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 12 6 10 6 10 5 8 4 6 4 6 3 4 3 4 2 q t q t q t q t q t q t q t q t 1 2 4 4 2 8 3 > ----- + - + 2 q t + q t + q t 2 2 t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n136

L11n135 L11n137