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L11n116 L11n118

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X17,21,18,20 X13,19,14,18 X19,15,20,14 X4,21,1,22 X10,5,11,6 X12,3,13,4 X22,11,5,12 X2,9,3,10 X8,15,9,16

Gauss Code:

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

Jones Polynomial:

- q-17/2 + 3q-15/2 - 5q-13/2 + 8q-11/2 - 10q-9/2 + 9q-7/2 - 10q-5/2 + 7q-3/2 - 5q-1/2 + 2q1/2

A2 (sl(3)) Invariant:

q-26 - q-24 - q-22 - 3q-18 + 2q-16 + 2q-14 + 3q-12 + 5q-10 + q-8 + 3q-6 - q-4 - q-2 + 1 - 2q2

HOMFLY-PT Polynomial:

az-1 + 3az + 2az3 - 4a3z-1 - 9a3z - 7a3z3 - 2a3z5 + 4a5z-1 + 3a5z - a5z3 - a5z5 - a7z-1 + a7z + a7z3

Kauffman Polynomial:

1 - 3z2 - az-1 + 4az - 5az3 - az5 + 4a2 - 10a2z2 + 9a2z4 - 5a2z6 - 4a3z-1 + 14a3z - 25a3z3 + 23a3z5 - 8a3z7 + 7a4 - 10a4z2 - 2a4z4 + 14a4z6 - 6a4z8 - 4a5z-1 + 13a5z - 31a5z3 + 30a5z5 - 4a5z7 - 2a5z9 + 4a6 + 3a6z2 - 28a6z4 + 32a6z6 - 9a6z8 - a7z-1 + 5a7z - 16a7z3 + 10a7z5 + 3a7z7 - 2a7z9 + a8 + 6a8z2 - 17a8z4 + 13a8z6 - 3a8z8 + 2a9z - 5a9z3 + 4a9z5 - a9z7

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 j = 2                   2 j = 0                 3   j = -2               5 3   j = -4             5 2     j = -6           4 5       j = -8         6 5         j = -10       3 5           j = -12     2 5             j = -14   1 3               j = -16   2                 j = -18 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, 117]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 117]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[17, 21, 18, 20], X[13, 19, 14, 18], > X[19, 15, 20, 14], X[4, 21, 1, 22], X[10, 5, 11, 6], X[12, 3, 13, 4], > X[22, 11, 5, 12], X[2, 9, 3, 10], X[8, 15, 9, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 8, -6}, {7, -1, 2, -11, 10, -7, 9, -8, -4, 5, 11, -2, -3, 4, > -5, 3, 6, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 3 5 8 10 9 10 7 5 -q + ----- - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q > 2 Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 -24 -22 3 2 2 3 5 -8 3 -4 -2 1 + q - q - q - --- + --- + --- + --- + --- + q + -- - q - q - 18 16 14 12 10 6 q q q q q q 2 > 2 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 117]][a, z]
Out[8]=	3 5 7 a 4 a 4 a a 3 5 7 3 3 3 - - ---- + ---- - -- + 3 a z - 9 a z + 3 a z + a z + 2 a z - 7 a z - z z z z 5 3 7 3 3 5 5 5 > a z + a z - 2 a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 117]][a, z]
Out[9]=	3 5 7 2 4 6 8 a 4 a 4 a a 3 1 + 4 a + 7 a + 4 a + a - - - ---- - ---- - -- + 4 a z + 14 a z + z z z z 5 7 9 2 2 2 4 2 6 2 > 13 a z + 5 a z + 2 a z - 3 z - 10 a z - 10 a z + 3 a z + 8 2 3 3 3 5 3 7 3 9 3 2 4 > 6 a z - 5 a z - 25 a z - 31 a z - 16 a z - 5 a z + 9 a z - 4 4 6 4 8 4 5 3 5 5 5 7 5 > 2 a z - 28 a z - 17 a z - a z + 23 a z + 30 a z + 10 a z + 9 5 2 6 4 6 6 6 8 6 3 7 5 7 > 4 a z - 5 a z + 14 a z + 32 a z + 13 a z - 8 a z - 4 a z + 7 7 9 7 4 8 6 8 8 8 5 9 7 9 > 3 a z - a z - 6 a z - 9 a z - 3 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 2 1 3 2 5 3 3 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 18 8 16 7 14 7 14 6 12 6 12 5 10 5 q q t q t q t q t q t q t q t 5 6 5 4 5 5 2 5 2 > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + 2 q t 10 4 8 4 8 3 6 3 6 2 4 2 4 2 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 L11n117

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