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L11n198 L11n200

Knotscape

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DrawMorseLink

PD Presentation:

X10,1,11,2 X12,3,13,4 X5,14,6,15 X16,7,17,8 X20,15,21,16 X13,18,14,19 X21,6,22,7 X17,22,18,9 X4,19,5,20 X2,9,3,10 X8,11,1,12

Gauss Code:

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

Jones Polynomial:

- q-19/2 + q-17/2 - q-15/2 + q-13/2 - 2q-11/2 + q-9/2 - q-7/2

A2 (sl(3)) Invariant:

- q-42 - q-36 + q-34 + q-32 + 2q-30 + q-28 + q-26 + q-24 + q-20 + q-18 + q-16 + q-12

HOMFLY-PT Polynomial:

- 5a7z - 10a7z3 - 6a7z5 - a7z7 - 2a9z-1 - 6a9z - 10a9z3 - 6a9z5 - a9z7 + 3a11z-1 + 8a11z + 6a11z3 + a11z5 - a13z-1 - a13z

Kauffman Polynomial:

5a7z - 10a7z3 + 6a7z5 - a7z7 - 5a8z4 + 5a8z6 - a8z8 + 2a9z-1 - 8a9z + 11a9z3 - 11a9z5 + 6a9z7 - a9z9 - 3a10 + 12a10z2 - 20a10z4 + 12a10z6 - 2a10z8 + 3a11z-1 - 14a11z + 22a11z3 - 17a11z5 + 7a11z7 - a11z9 - 3a12 + 13a12z2 - 15a12z4 + 7a12z6 - a12z8 + a13z-1 - a13z + a13z3 - a14 + a14z2

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             1     j = -12         1 1 1     j = -14       1 2 1       j = -16       1 1         j = -18     2 2           j = -20 1                 j = -22 1 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, 199]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 199]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 3, 13, 4], X[5, 14, 6, 15], X[16, 7, 17, 8], > X[20, 15, 21, 16], X[13, 18, 14, 19], X[21, 6, 22, 7], X[17, 22, 18, 9], > X[4, 19, 5, 20], X[2, 9, 3, 10], X[8, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -9, -3, 7, 4, -11}, > {10, -1, 11, -2, -6, 3, 5, -4, -8, 6, 9, -5, -7, 8}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) -(17/2) -(15/2) -(13/2) 2 -(9/2) -(7/2) -q + q - q + q - ----- + q - q 11/2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 -36 -34 -32 2 -28 -26 -24 -20 -18 -16 -q - q + q + q + --- + q + q + q + q + q + q + 30 q -12 > q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 199]][a, z]
Out[8]=	9 11 13 -2 a 3 a a 7 9 11 13 7 3 9 3 ----- + ----- - --- - 5 a z - 6 a z + 8 a z - a z - 10 a z - 10 a z + z z z 11 3 7 5 9 5 11 5 7 7 9 7 > 6 a z - 6 a z - 6 a z + a z - a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 199]][a, z]
Out[9]=	9 11 13 10 12 14 2 a 3 a a 7 9 11 -3 a - 3 a - a + ---- + ----- + --- + 5 a z - 8 a z - 14 a z - z z z 13 10 2 12 2 14 2 7 3 9 3 11 3 > a z + 12 a z + 13 a z + a z - 10 a z + 11 a z + 22 a z + 13 3 8 4 10 4 12 4 7 5 9 5 11 5 > a z - 5 a z - 20 a z - 15 a z + 6 a z - 11 a z - 17 a z + 8 6 10 6 12 6 7 7 9 7 11 7 8 8 > 5 a z + 12 a z + 7 a z - a z + 6 a z + 7 a z - a z - 10 8 12 8 9 9 11 9 > 2 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-8 -6 1 1 1 2 2 1 1 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 22 8 20 8 22 7 18 6 18 5 16 5 14 5 q t q t q t q t q t q t q t 1 2 1 1 1 1 1 1 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ---- 16 4 14 4 12 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 L11n199

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