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L11n177 L11n179

Knotscape

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DrawMorseLink

PD Presentation:

X8192 X18,9,19,10 X13,21,14,20 X3,10,4,11 X5,14,6,15 X7,16,8,17 X15,22,16,7 X11,4,12,5 X19,13,20,12 X21,1,22,6 X2,18,3,17

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

- q-22 + q-20 - q-18 + q-16 + q-14 - q-12 + 3q-10 + 3q-6 + q-4 + 2 - q2 + q4

HOMFLY-PT Polynomial:

- az-1 - 3az - 3az3 - az5 + a3z-1 + 4a3z + 8a3z3 + 5a3z5 + a3z7 - 2a5z - 3a5z3 - a5z5

Kauffman Polynomial:

- a-1z3 + z2 - 3z4 + az-1 - 4az + 7az3 - 6az5 - a2 + a2z2 - a2z6 - a2z8 + a3z-1 - 6a3z + 12a3z3 - 6a3z5 + a3z7 - a3z9 + a4z2 - 3a4z4 + 8a4z6 - 4a4z8 - a5z - 4a5z3 + 10a5z5 - 2a5z7 - a5z9 - a6z2 - 3a6z4 + 8a6z6 - 3a6z8 + a7z - 8a7z3 + 10a7z5 - 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                   1 j = 2                 2   j = 0               4 1   j = -2             4 3     j = -4           5 3       j = -6         4 5         j = -8       3 4           j = -10     2 4             j = -12   1 3               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, 178]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 178]]
Out[4]=	PD[X[8, 1, 9, 2], X[18, 9, 19, 10], X[13, 21, 14, 20], X[3, 10, 4, 11], > X[5, 14, 6, 15], X[7, 16, 8, 17], X[15, 22, 16, 7], X[11, 4, 12, 5], > X[19, 13, 20, 12], X[21, 1, 22, 6], X[2, 18, 3, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, -4, 8, -5, 10}, > {-6, -1, 2, 4, -8, 9, -3, 5, -7, 6, 11, -2, -9, 3, -10, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 3 5 7 8 9 7 6 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 3 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -20 -18 -16 -14 -12 3 3 -4 2 4 2 - q + q - q + q + q - q + --- + -- + q - q + q 10 6 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 178]][a, z]
Out[8]=	3 a a 3 5 3 3 3 5 3 5 -(-) + -- - 3 a z + 4 a z - 2 a z - 3 a z + 8 a z - 3 a z - a z + z z 3 5 5 5 3 7 > 5 a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 178]][a, z]
Out[9]=	3 2 a a 3 5 7 2 2 2 4 2 6 2 -a + - + -- - 4 a z - 6 a z - a z + a z + z + a z + a z - a z - z z 3 8 2 z 3 3 3 5 3 7 3 4 4 4 > 2 a z - -- + 7 a z + 12 a z - 4 a z - 8 a z - 3 z - 3 a z - a 6 4 8 4 5 3 5 5 5 7 5 2 6 > 3 a z + 3 a z - 6 a z - 6 a z + 10 a z + 10 a z - a z + 4 6 6 6 8 6 3 7 5 7 7 7 2 8 4 8 > 8 a z + 8 a z - a z + a z - 2 a z - 3 a z - a z - 4 a z - 6 8 3 9 5 9 > 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 2 1 3 2 4 3 4 4 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 5 3 4 2 4 2 > ----- + ----- + ----- + ---- + ---- + t + 2 q t + 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 L11n178

L11n177 L11n179