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L11n171 L11n173

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DrawMorseLink

PD Presentation:

X8192 X3,10,4,11 X12,7,13,8 X22,15,7,16 X14,6,15,5 X6,14,1,13 X16,21,17,22 X9,18,10,19 X20,11,21,12 X4,18,5,17 X19,3,20,2

Gauss Code:

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

Jones Polynomial:

q-15/2 - 3q-13/2 + 6q-11/2 - 8q-9/2 + 10q-7/2 - 11q-5/2 + 9q-3/2 - 8q-1/2 + 4q1/2 - 2q3/2

A2 (sl(3)) Invariant:

- q-24 + 2q-20 - 2q-18 - q-14 - 4q-12 + q-10 + 4q-6 + 3q-4 + q-2 + 4 - q2 + q4 + 2q6

HOMFLY-PT Polynomial:

- a-1z-1 - 2a-1z + 3az + 3az3 + 2a3z-1 - a3z3 - a3z5 - a5z-1 + a5z + 2a5z3 - a7z

Kauffman Polynomial:

- a-1z-1 + 4a-1z - 3a-1z3 + 1 + z2 - 2z4 - z6 + 3az - 8az3 + 5az5 - 3az7 - 3a2 + 11a2z2 - 18a2z4 + 11a2z6 - 4a2z8 + 2a3z-1 - 2a3z - 4a3z3 + 4a3z5 + a3z7 - 2a3z9 - 5a4 + 18a4z2 - 29a4z4 + 25a4z6 - 8a4z8 + a5z-1 - 5a5z3 + 8a5z5 + a5z7 - 2a5z9 - 2a6 + 6a6z2 - 10a6z4 + 12a6z6 - 4a6z8 + a7z - 6a7z3 + 9a7z5 - 3a7z7 - 2a8z2 + 3a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 4                   2 j = 2                 3 1 j = 0               5 1   j = -2             5 4     j = -4           6 4       j = -6         4 5         j = -8       4 6           j = -10     2 4             j = -12   1 4               j = -14   2                 j = -16 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, 172]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 172]]
Out[4]=	PD[X[8, 1, 9, 2], X[3, 10, 4, 11], X[12, 7, 13, 8], X[22, 15, 7, 16], > X[14, 6, 15, 5], X[6, 14, 1, 13], X[16, 21, 17, 22], X[9, 18, 10, 19], > X[20, 11, 21, 12], X[4, 18, 5, 17], X[19, 3, 20, 2]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 11, -2, -10, 5, -6}, > {3, -1, -8, 2, 9, -3, 6, -5, 4, -7, 10, 8, -11, -9, 7, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 3 6 8 10 11 9 8 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 4 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 > 2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 2 2 -14 4 -10 4 3 -2 2 4 6 4 - q + --- - --- - q - --- + q + -- + -- + q - q + q + 2 q 20 18 12 6 4 q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 172]][a, z]
Out[8]=	3 5 1 2 a a 2 z 5 7 3 3 3 5 3 -(---) + ---- - -- - --- + 3 a z + a z - a z + 3 a z - a z + 2 a z - a z z z a 3 5 > a z
In[9]:=	Kauffman[Link[11, NonAlternating, 172]][a, z]
Out[9]=	3 5 2 4 6 1 2 a a 4 z 3 7 2 1 - 3 a - 5 a - 2 a - --- + ---- + -- + --- + 3 a z - 2 a z + a z + z + a z z z a 3 2 2 4 2 6 2 8 2 3 z 3 3 3 > 11 a z + 18 a z + 6 a z - 2 a z - ---- - 8 a z - 4 a z - a 5 3 7 3 4 2 4 4 4 6 4 8 4 > 5 a z - 6 a z - 2 z - 18 a z - 29 a z - 10 a z + 3 a z + 5 3 5 5 5 7 5 6 2 6 4 6 > 5 a z + 4 a z + 8 a z + 9 a z - z + 11 a z + 25 a z + 6 6 8 6 7 3 7 5 7 7 7 2 8 4 8 > 12 a z - a z - 3 a z + a z + a z - 3 a z - 4 a z - 8 a z - 6 8 3 9 5 9 > 4 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 1 2 1 4 2 4 4 6 5 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 4 5 6 4 5 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + t + 3 q t + q t + 2 q t 6 3 6 2 4 2 4 2 q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n172

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