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L11n170 L11n172

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DrawMorseLink

PD Presentation:

X8192 X11,21,12,20 X10,4,11,3 X2,17,3,18 X14,5,15,6 X6718 X16,10,17,9 X13,19,14,18 X22,16,7,15 X19,13,20,12 X4,22,5,21

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 4q-5/2 - 8q-3/2 + 10q-1/2 - 14q1/2 + 13q3/2 - 12q5/2 + 9q7/2 - 5q9/2 + 2q11/2

A2 (sl(3)) Invariant:

q-10 - 2q-8 + 2q-6 + q-2 + 6 + 5q4 - q6 + q8 + q10 - 3q12 + q14 - 2q16 - q18

HOMFLY-PT Polynomial:

a-5z-1 + a-5z - 2a-3z-1 - 2a-3z + a-3z3 + a-3z5 - a-1z - 5a-1z3 - 4a-1z5 - a-1z7 + az-1 + az + 2az3 + az5

Kauffman Polynomial:

- 2a-6 + 5a-6z2 - 3a-6z4 + a-5z-1 - a-5z + 5a-5z3 - 3a-5z5 - a-5z7 - 5a-4 + 15a-4z2 - 14a-4z4 + 6a-4z6 - 3a-4z8 + 2a-3z-1 - 3a-3z + a-3z3 - 2a-3z9 - 3a-2 + 12a-2z2 - 22a-2z4 + 20a-2z6 - 9a-2z8 - 12a-1z3 + 18a-1z5 - 6a-1z7 - 2a-1z9 + 1 + 2z2 - 5z4 + 10z6 - 6z8 - az-1 + 2az - 7az3 + 14az5 - 7az7 + 6a2z4 - 4a2z6 + a3z3 - a3z5

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 12                   2 j = 10                 3   j = 8               6 2   j = 6             6 3     j = 4           7 6       j = 2         7 6         j = 0       4 8           j = -2     4 6             j = -4   1 5               j = -6   3                 j = -8 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, 171]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 171]]
Out[4]=	PD[X[8, 1, 9, 2], X[11, 21, 12, 20], X[10, 4, 11, 3], X[2, 17, 3, 18], > X[14, 5, 15, 6], X[6, 7, 1, 8], X[16, 10, 17, 9], X[13, 19, 14, 18], > X[22, 16, 7, 15], X[19, 13, 20, 12], X[4, 22, 5, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -11, 5, -6}, > {6, -1, 7, -3, -2, 10, -8, -5, 9, -7, 4, 8, -10, 2, 11, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 4 8 10 3/2 5/2 7/2 -q + ---- - ---- + ------- - 14 Sqrt[q] + 13 q - 12 q + 9 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 > 5 q + 2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 2 2 -2 4 6 8 10 12 14 16 18 6 + q - -- + -- + q + 5 q - q + q + q - 3 q + q - 2 q - q 8 6 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 171]][a, z]
Out[8]=	3 3 5 5 1 2 a z 2 z z z 5 z 3 z 4 z 5 ---- - ---- + - + -- - --- - - + a z + -- - ---- + 2 a z + -- - ---- + a z - 5 3 z 5 3 a 3 a 3 a a z a z a a a a 7 z > -- a
In[9]:=	Kauffman[Link[11, NonAlternating, 171]][a, z]
Out[9]=	2 2 2 5 3 1 2 a z 3 z 2 5 z 15 z 1 - -- - -- - -- + ---- + ---- - - - -- - --- + 2 a z + 2 z + ---- + ----- + 6 4 2 5 3 z 5 3 6 4 a a a a z a z a a a a 2 3 3 3 4 4 4 12 z 5 z z 12 z 3 3 3 4 3 z 14 z 22 z > ----- + ---- + -- - ----- - 7 a z + a z - 5 z - ---- - ----- - ----- + 2 5 3 a 6 4 2 a a a a a a 5 5 6 6 2 4 3 z 18 z 5 3 5 6 6 z 20 z 2 6 > 6 a z - ---- + ----- + 14 a z - a z + 10 z + ---- + ----- - 4 a z - 5 a 4 2 a a a 7 7 8 8 9 9 z 6 z 7 8 3 z 9 z 2 z 2 z > -- - ---- - 7 a z - 6 z - ---- - ---- - ---- - ---- 5 a 4 2 3 a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 3 1 5 4 4 6 2 4 8 + 7 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 6 q t + 7 q t + 8 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t 4 2 6 2 6 3 8 3 8 4 10 4 12 5 > 6 q t + 6 q t + 3 q t + 6 q t + 2 q t + 3 q t + 2 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n171

L11n170 L11n172