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L11n228 L11n230

Knotscape

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DrawMorseLink

PD Presentation:

X10,1,11,2 X2,11,3,12 X12,3,13,4 X20,5,21,6 X7,14,8,15 X13,16,14,17 X15,8,16,1 X22,17,9,18 X18,21,19,22 X6,9,7,10 X4,19,5,20

Gauss Code:

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

Jones Polynomial:

- 2q-23/2 + 3q-21/2 - 3q-19/2 + 4q-17/2 - 4q-15/2 + 3q-13/2 - 3q-11/2 + q-9/2 - q-7/2

A2 (sl(3)) Invariant:

- q-42 + q-40 + q-38 + q-36 + q-34 - q-32 - 2q-28 + q-26 + q-24 + q-22 + 2q-20 + q-18 + 2q-16 + q-12

HOMFLY-PT Polynomial:

- a7z-1 - 7a7z - 11a7z3 - 6a7z5 - a7z7 + a9z-1 - a9z - 6a9z3 - 5a9z5 - a9z7 + 5a11z + 5a11z3 + a11z5 - a13z

Kauffman Polynomial:

- a7z-1 + 7a7z - 11a7z3 + 6a7z5 - a7z7 + a8 - 3a8z2 - 2a8z4 + 4a8z6 - a8z8 - a9z-1 + a9z - 3a9z5 + 4a9z7 - a9z9 + 8a10z2 - 18a10z4 + 14a10z6 - 3a10z8 - 5a11z + 9a11z3 - 6a11z5 + 4a11z7 - a11z9 + 11a12z2 - 17a12z4 + 10a12z6 - 2a12z8 - a13z - 2a13z3 + 3a13z5 - a13z7 - a14z4 - 2a15z

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -6                 1 j = -8               1 1 j = -10             2     j = -12           1 1     j = -14         3 2       j = -16       1 1         j = -18     2 3           j = -20   1 1             j = -22 1 2               j = -24 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, 229]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 229]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[20, 5, 21, 6], > X[7, 14, 8, 15], X[13, 16, 14, 17], X[15, 8, 16, 1], X[22, 17, 9, 18], > X[18, 21, 19, 22], X[6, 9, 7, 10], X[4, 19, 5, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -11, 4, -10, -5, 7}, > {10, -1, 2, -3, -6, 5, -7, 6, 8, -9, 11, -4, 9, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-2 3 3 4 4 3 3 -(9/2) -(7/2) ----- + ----- - ----- + ----- - ----- + ----- - ----- + q - q 23/2 21/2 19/2 17/2 15/2 13/2 11/2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 -40 -38 -36 -34 -32 2 -26 -24 -22 2 -q + q + q + q + q - q - --- + q + q + q + --- + 28 20 q q -18 2 -12 > q + --- + q 16 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 229]][a, z]
Out[8]=	7 9 a a 7 9 11 13 7 3 9 3 11 3 -(--) + -- - 7 a z - a z + 5 a z - a z - 11 a z - 6 a z + 5 a z - z z 7 5 9 5 11 5 7 7 9 7 > 6 a z - 5 a z + a z - a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 229]][a, z]
Out[9]=	7 9 8 a a 7 9 11 13 15 8 2 10 2 a - -- - -- + 7 a z + a z - 5 a z - a z - 2 a z - 3 a z + 8 a z + z z 12 2 7 3 11 3 13 3 8 4 10 4 > 11 a z - 11 a z + 9 a z - 2 a z - 2 a z - 18 a z - 12 4 14 4 7 5 9 5 11 5 13 5 8 6 > 17 a z - a z + 6 a z - 3 a z - 6 a z + 3 a z + 4 a z + 10 6 12 6 7 7 9 7 11 7 13 7 8 8 > 14 a z + 10 a z - a z + 4 a z + 4 a z - a z - a z - 10 8 12 8 9 9 11 9 > 3 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-8 -6 2 1 2 1 1 2 3 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 24 8 22 8 22 7 20 7 20 6 18 6 18 5 q t q t q t q t q t q t q t 1 1 3 2 1 1 2 1 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ---- 16 5 16 4 14 4 14 3 12 3 12 2 10 2 8 q t q t q t q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n229

L11n228 L11n230