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L11n180 L11n182

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DrawMorseLink

PD Presentation:

X8192 X3,10,4,11 X5,14,6,15 X16,8,17,7 X22,18,7,17 X15,13,16,12 X20,10,21,9 X11,19,12,18 X13,6,14,1 X19,4,20,5 X2,21,3,22

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

q-18 + 2q-16 + 3q-14 + 2q-12 + 2q-10 + 2q-8 - q-6 - q-2 - q4 - q8 + q12

HOMFLY-PT Polynomial:

- a-3z + a-1z + a-1z3 + az-1 + az + az3 - 3a3z-1 - 5a3z - a3z3 + 2a5z-1 + a5z

Kauffman Polynomial:

- a-3z + 3a-3z3 - a-3z5 - 5a-2z2 + 7a-2z4 - 2a-2z6 - a-1z + a-1z3 + 2a-1z5 - a-1z7 + 1 - 3z2 + 5z4 - 2z6 - az-1 + 3az3 - 2az5 + 3a2 - 4a2z2 - 3a3z-1 + 9a3z - 9a3z3 + 2a3z5 + 3a4 - 6a4z2 + 2a4z4 - 2a5z-1 + 9a5z - 14a5z3 + 7a5z5 - a5z7

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                     1 j = 6                   1   j = 4                 1 1   j = 2               2 1     j = 0           1 2 1       j = -2           2 3         j = -4       1 2 1           j = -6         2             j = -8     1 1               j = -10 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, 181]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 181]]
Out[4]=	PD[X[8, 1, 9, 2], X[3, 10, 4, 11], X[5, 14, 6, 15], X[16, 8, 17, 7], > X[22, 18, 7, 17], X[15, 13, 16, 12], X[20, 10, 21, 9], X[11, 19, 12, 18], > X[13, 6, 14, 1], X[19, 4, 20, 5], X[2, 21, 3, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, -2, 10, -3, 9}, > {4, -1, 7, 2, -8, 6, -9, 3, -6, -4, 5, 8, -10, -7, 11, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 2 2 3 3/2 5/2 7/2 -q - ---- + ---- - ------- + 3 Sqrt[q] - 2 q + 2 q - q 5/2 3/2 Sqrt[q] q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 2 3 2 2 2 -6 -2 4 8 12 q + --- + --- + --- + --- + -- - q - q - q - q + q 16 14 12 10 8 q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 181]][a, z]
Out[8]=	3 5 3 a 3 a 2 a z z 3 5 z 3 3 3 - - ---- + ---- - -- + - + a z - 5 a z + a z + -- + a z - a z z z z 3 a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 181]][a, z]
Out[9]=	3 5 2 2 4 a 3 a 2 a z z 3 5 2 5 z 1 + 3 a + 3 a - - - ---- - ---- - -- - - + 9 a z + 9 a z - 3 z - ---- - z z z 3 a 2 a a 3 3 4 2 2 4 2 3 z z 3 3 3 5 3 4 7 z > 4 a z - 6 a z + ---- + -- + 3 a z - 9 a z - 14 a z + 5 z + ---- + 3 a 2 a a 5 5 6 7 4 4 z 2 z 5 3 5 5 5 6 2 z z 5 7 > 2 a z - -- + ---- - 2 a z + 2 a z + 7 a z - 2 z - ---- - -- - a z 3 a 2 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 1 1 1 2 2 1 1 2 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + - + ---- + 2 12 6 10 6 8 4 8 3 4 3 6 2 4 2 t 4 q q t q t q t q t q t q t q t q t 2 2 2 2 4 2 4 3 6 3 8 4 > ---- + t + 2 q t + q t + q t + q t + q t + q t 2 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n181

L11n180 L11n182