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L11n160 L11n162

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DrawMorseLink

PD Presentation:

X8192 X16,7,17,8 X10,4,11,3 X2,15,3,16 X14,10,15,9 X18,11,19,12 X5,13,6,12 X6,21,1,22 X13,20,14,21 X22,17,7,18 X19,4,20,5

Gauss Code:

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

Jones Polynomial:

- q-17/2 + 3q-15/2 - 4q-13/2 + 6q-11/2 - 7q-9/2 + 6q-7/2 - 7q-5/2 + 4q-3/2 - 3q-1/2 + q1/2

A2 (sl(3)) Invariant:

q-26 - q-24 - q-22 - q-20 - 3q-18 + q-16 + q-14 + 3q-12 + 4q-10 + 2q-8 + 3q-6 + 1 - q2

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- z2 + 2az - 3az3 - a2z6 - 2a3z-1 + 6a3z - 10a3z3 + 9a3z5 - 3a3z7 + 3a4 + 4a4z2 - 16a4z4 + 15a4z6 - 4a4z8 - 3a5z-1 + 5a5z - 10a5z3 + 6a5z5 + 4a5z7 - 2a5z9 + 3a6 + 9a6z2 - 34a6z4 + 30a6z6 - 7a6z8 - a7z-1 + 2a7z - 7a7z3 + a7z5 + 6a7z7 - 2a7z9 + a8 + 6a8z2 - 18a8z4 + 14a8z6 - 3a8z8 + a9z - 4a9z3 + 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                   1 j = 0                 2   j = -2               3 2   j = -4             4 1     j = -6           3 4       j = -8         4 3         j = -10       2 3           j = -12     2 4             j = -14   1 2               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, 161]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 161]]
Out[4]=	PD[X[8, 1, 9, 2], X[16, 7, 17, 8], X[10, 4, 11, 3], X[2, 15, 3, 16], > X[14, 10, 15, 9], X[18, 11, 19, 12], X[5, 13, 6, 12], X[6, 21, 1, 22], > X[13, 20, 14, 21], X[22, 17, 7, 18], X[19, 4, 20, 5]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, 11, -7, -8}, > {2, -1, 5, -3, 6, 7, -9, -5, 4, -2, 10, -6, -11, 9, 8, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 3 4 6 7 6 7 4 3 -q + ----- - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q > Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 -24 -22 -20 3 -16 -14 3 4 2 3 2 1 + q - q - q - q - --- + q + q + --- + --- + -- + -- - q 18 12 10 8 6 q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 161]][a, z]
Out[8]=	3 5 7 -2 a 3 a a 3 5 7 3 3 3 5 3 ----- + ---- - -- + a z - 4 a z + a z + a z + a z - 3 a z - 2 a z + z z z 7 3 3 5 5 5 > a z - a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 161]][a, z]
Out[9]=	3 5 7 4 6 8 2 a 3 a a 3 5 7 9 3 a + 3 a + a - ---- - ---- - -- + 2 a z + 6 a z + 5 a z + 2 a z + a z - z z z 2 4 2 6 2 8 2 3 3 3 5 3 7 3 > z + 4 a z + 9 a z + 6 a z - 3 a z - 10 a z - 10 a z - 7 a z - 9 3 4 4 6 4 8 4 3 5 5 5 7 5 > 4 a z - 16 a z - 34 a z - 18 a z + 9 a z + 6 a z + a z + 9 5 2 6 4 6 6 6 8 6 3 7 5 7 > 4 a z - a z + 15 a z + 30 a z + 14 a z - 3 a z + 4 a z + 7 7 9 7 4 8 6 8 8 8 5 9 7 9 > 6 a z - a z - 4 a z - 7 a z - 3 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 2 2 4 2 2 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 3 4 3 3 4 4 1 3 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 L11n161

L11n160 L11n162