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L11n139 L11n141

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DrawMorseLink

PD Presentation:

X8192 X18,11,19,12 X3,10,4,11 X17,3,18,2 X12,5,13,6 X6718 X9,16,10,17 X15,20,16,21 X13,22,14,7 X21,14,22,15 X4,20,5,19

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:

2q-17/2 - 3q-15/2 + 4q-13/2 - 6q-11/2 + 5q-9/2 - 5q-7/2 + 3q-5/2 - 2q-3/2

A2 (sl(3)) Invariant:

- q-28 - 3q-26 - q-24 - q-22 + q-20 + 4q-18 + 2q-16 + 3q-14 + q-10 + 2q-8 + 2q-4

HOMFLY-PT Polynomial:

- a3z-1 - 4a3z - 2a3z3 + 3a5z + 3a5z3 + a5z5 + 2a7z-1 - a7z3 - a9z-1

Kauffman Polynomial:

- a3z-1 + 5a3z - 3a3z3 + a4 - a4z6 + 5a5z - 11a5z3 + 6a5z5 - 2a5z7 - 3a6 + 6a6z2 - 6a6z4 + 2a6z6 - a6z8 + 2a7z-1 - 2a7z - 6a7z3 + 7a7z5 - 3a7z7 - 5a8 + 13a8z2 - 9a8z4 + 3a8z6 - a8z8 + a9z-1 - 2a9z + 2a9z3 + a9z5 - a9z7 - 2a10 + 7a10z2 - 3a10z4

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -2               2 j = -4             2 1 j = -6           3 1   j = -8         3 3     j = -10       3 2       j = -12     1 3         j = -14   2 3           j = -16   1             j = -18 2

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, 140]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 140]]
Out[4]=	PD[X[8, 1, 9, 2], X[18, 11, 19, 12], X[3, 10, 4, 11], X[17, 3, 18, 2], > 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[4, 20, 5, 19]]
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]=	2 3 4 6 5 5 3 2 ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 3 -24 -22 -20 4 2 3 -10 2 2 -q - --- - q - q + q + --- + --- + --- + q + -- + -- 26 18 16 14 8 4 q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 140]][a, z]
Out[8]=	3 7 9 a 2 a a 3 5 3 3 5 3 7 3 5 5 -(--) + ---- - -- - 4 a z + 3 a z - 2 a z + 3 a z - a z + a z z z z
In[9]:=	Kauffman[Link[11, NonAlternating, 140]][a, z]
Out[9]=	3 7 9 4 6 8 10 a 2 a a 3 5 7 9 a - 3 a - 5 a - 2 a - -- + ---- + -- + 5 a z + 5 a z - 2 a z - 2 a z + z z z 6 2 8 2 10 2 3 3 5 3 7 3 9 3 > 6 a z + 13 a z + 7 a z - 3 a z - 11 a z - 6 a z + 2 a z - 6 4 8 4 10 4 5 5 7 5 9 5 4 6 > 6 a z - 9 a z - 3 a z + 6 a z + 7 a z + a z - a z + 6 6 8 6 5 7 7 7 9 7 6 8 8 8 > 2 a z + 3 a z - 2 a z - 3 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 2 2 1 2 3 1 3 3 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 18 7 16 6 14 6 14 5 12 5 12 4 10 4 q q t q t q t q t q t q t q t 2 3 3 3 1 2 > ------ + ----- + ----- + ----- + ---- + ---- 10 3 8 3 8 2 6 2 6 4 q t q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n140

L11n139 L11n141