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L11n138 L11n140

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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 X16,10,17,9 X20,16,21,15 X22,14,7,13 X14,22,15,21 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-3/2 - q1/2 - q7/2 + q9/2 - q11/2 + q13/2

A2 (sl(3)) Invariant:

q-6 + q-4 + 2q-2 + 3 + 2q2 + 2q4 + q8 - q16 - q18 - q20

HOMFLY-PT Polynomial:

a-5z-1 + 3a-5z + a-5z3 - 2a-3z-1 - 6a-3z - 5a-3z3 - a-3z5 + a-1z + az-1 + az

Kauffman Polynomial:

- 2a-6 + 11a-6z2 - 15a-6z4 + 7a-6z6 - a-6z8 + a-5z-1 - 4a-5z + 12a-5z3 - 15a-5z5 + 7a-5z7 - a-5z9 - 5a-4 + 21a-4z2 - 29a-4z4 + 14a-4z6 - 2a-4z8 + 2a-3z-1 - 8a-3z + 14a-3z3 - 15a-3z5 + 7a-3z7 - a-3z9 - 3a-2 + 10a-2z2 - 14a-2z4 + 7a-2z6 - a-2z8 - a-1z + a-1z3 + 1 - az-1 + 3az - az3

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 14                   1 j = 12                     j = 10               1 1   j = 8           1 1       j = 6             1       j = 4       1 2 1         j = 2     1               j = 0     2 1             j = -2 1                   j = -4 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, 139]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 139]]
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[16, 10, 17, 9], X[20, 16, 21, 15], > X[22, 14, 7, 13], X[14, 22, 15, 21], 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]=	-(3/2) 7/2 9/2 11/2 13/2 -q - Sqrt[q] - q + q - q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 -4 2 2 4 8 16 18 20 3 + q + q + -- + 2 q + 2 q + q - q - q - q 2 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 139]][a, z]
Out[8]=	3 3 5 1 2 a 3 z 6 z z z 5 z z ---- - ---- + - + --- - --- + - + a z + -- - ---- - -- 5 3 z 5 3 a 5 3 3 a z a z a a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 139]][a, z]
Out[9]=	2 2 2 5 3 1 2 a 4 z 8 z z 11 z 21 z 1 - -- - -- - -- + ---- + ---- - - - --- - --- - - + 3 a z + ----- + ----- + 6 4 2 5 3 z 5 3 a 6 4 a a a a z a z a a a a 2 3 3 3 4 4 4 5 5 10 z 12 z 14 z z 3 15 z 29 z 14 z 15 z 15 z > ----- + ----- + ----- + -- - a z - ----- - ----- - ----- - ----- - ----- + 2 5 3 a 6 4 2 5 3 a a a a a a a a 6 6 6 7 7 8 8 8 9 9 7 z 14 z 7 z 7 z 7 z z 2 z z z z > ---- + ----- + ---- + ---- + ---- - -- - ---- - -- - -- - -- 6 4 2 5 3 6 4 2 5 3 a a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 4 4 2 4 3 8 3 6 4 8 4 2 + q + ----- + ----- + t + q t + 2 q t + q t + q t + q t + q t + 4 2 2 2 q t q t 10 5 10 6 14 7 > q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n139

L11n138 L11n140