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L11n155 L11n157

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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X11,21,12,20 X5,12,6,13 X19,4,20,5 X14,18,15,17 X16,8,17,7 X22,16,7,15 X18,14,19,13 X6,21,1,22

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 2q-11/2 - 5q-9/2 + 7q-7/2 - 9q-5/2 + 9q-3/2 - 9q-1/2 + 6q1/2 - 4q3/2 + 2q5/2

A2 (sl(3)) Invariant:

q-20 + q-16 + 3q-14 + 3q-10 + q-8 + 2q-4 - q-2 + 3 - q2 - q4 - 2q8

HOMFLY-PT Polynomial:

a-1z-1 + 4a-1z + 2a-1z3 - 2az-1 - 8az - 7az3 - 2az5 - a3z - 2a3z3 - a3z5 + a5z-1 + 2a5z + a5z3

Kauffman Polynomial:

- 2a-2 + 6a-2z2 - 3a-2z4 + a-1z-1 - 4a-1z + 5a-1z3 - a-1z5 - a-1z7 - 5 + 18z2 - 16z4 + 6z6 - 2z8 + 2az-1 - 10az + 13az3 - 7az5 + az7 - az9 - 3a2 + 10a2z2 - 14a2z4 + 9a2z6 - 4a2z8 + a3z - 2a3z3 + 2a3z5 - a3z7 - a3z9 + a4 - 3a4z2 + 3a4z4 + a4z6 - 2a4z8 - a5z-1 + 5a5z - 7a5z3 + 7a5z5 - 3a5z7 - a6z2 + 4a6z4 - 2a6z6 - 2a7z + 3a7z3 - a7z5

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 6                   2 j = 4                 2   j = 2               4 2   j = 0             5 2     j = -2           5 5       j = -4         4 4         j = -6       3 5           j = -8     2 4             j = -10     3               j = -12 1 2                 j = -14 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, 156]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 156]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[11, 21, 12, 20], > X[5, 12, 6, 13], X[19, 4, 20, 5], X[14, 18, 15, 17], X[16, 8, 17, 7], > X[22, 16, 7, 15], X[18, 14, 19, 13], X[6, 21, 1, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, 6, -5, -11}, > {8, -1, 2, -3, -4, 5, 10, -7, 9, -8, 7, -10, -6, 4, 11, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 2 5 7 9 9 9 3/2 -q + ----- - ---- + ---- - ---- + ---- - ------- + 6 Sqrt[q] - 4 q + 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 5/2 > 2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -16 3 3 -8 2 -2 2 4 8 3 + q + q + --- + --- + q + -- - q - q - q - 2 q 14 10 4 q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 156]][a, z]
Out[8]=	5 3 1 2 a a 4 z 3 5 2 z 3 3 3 --- - --- + -- + --- - 8 a z - a z + 2 a z + ---- - 7 a z - 2 a z + a z z z a a 5 3 5 3 5 > a z - 2 a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 156]][a, z]
Out[9]=	5 2 2 4 1 2 a a 4 z 3 5 7 -5 - -- - 3 a + a + --- + --- - -- - --- - 10 a z + a z + 5 a z - 2 a z + 2 a z z z a a 2 3 2 6 z 2 2 4 2 6 2 5 z 3 3 3 > 18 z + ---- + 10 a z - 3 a z - a z + ---- + 13 a z - 2 a z - 2 a a 4 5 5 3 7 3 4 3 z 2 4 4 4 6 4 z > 7 a z + 3 a z - 16 z - ---- - 14 a z + 3 a z + 4 a z - -- - 2 a a 5 3 5 5 5 7 5 6 2 6 4 6 6 6 > 7 a z + 2 a z + 7 a z - a z + 6 z + 9 a z + a z - 2 a z - 7 z 7 3 7 5 7 8 2 8 4 8 9 3 9 > -- + a z - a z - 3 a z - 2 z - 4 a z - 2 a z - a z - a z a
In[10]:=	Kh[L][q, t]
Out[10]=	5 1 1 2 3 2 4 3 5 5 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 6 12 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 4 4 5 2 2 2 4 2 6 3 > ----- + ---- + ---- + 2 t + 4 q t + 2 q t + 2 q t + 2 q t 4 2 4 2 q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n156

L11n155 L11n157