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L11a167 L11a169

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DrawMorseLink

PD Presentation:

X8192 X18,11,19,12 X10,4,11,3 X2,17,3,18 X14,5,15,6 X6718 X16,10,17,9 X20,13,21,14 X12,19,13,20 X22,16,7,15 X4,22,5,21

Gauss Code:

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

Jones Polynomial:

q-17/2 - 4q-15/2 + 8q-13/2 - 13q-11/2 + 18q-9/2 - 20q-7/2 + 19q-5/2 - 18q-3/2 + 12q-1/2 - 8q1/2 + 4q3/2 - q5/2

A2 (sl(3)) Invariant:

- q-24 + 2q-22 - 2q-20 + 3q-18 - q-16 - 2q-14 + 2q-12 - 4q-10 + 7q-8 + 4q-4 + 2q-2 - 2 + 2q2 - 2q4 + q6

HOMFLY-PT Polynomial:

- az-1 - az - 4az3 - 4az5 - az7 + a3z-1 + 8a3z3 + 12a3z5 + 6a3z7 + a3z9 - 4a5z3 - 4a5z5 - a5z7

Kauffman Polynomial:

- 2a-1z3 + 3a-1z5 - a-1z7 + 2z2 - 13z4 + 14z6 - 4z8 + az-1 - az + 8az3 - 22az5 + 21az7 - 6az9 - a2 + 4a2z2 - 23a2z4 + 22a2z6 - 3a2z10 + a3z-1 - a3z + 14a3z3 - 38a3z5 + 42a3z7 - 14a3z9 - 18a4z4 + 30a4z6 - 7a4z8 - 3a4z10 - 2a5z3 + 4a5z5 + 9a5z7 - 8a5z9 - 4a6z2 + 14a6z6 - 11a6z8 - 4a7z3 + 13a7z5 - 11a7z7 - 2a8z2 + 7a8z4 - 8a8z6 + 2a9z3 - 4a9z5 - a10z4

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 6                       1 j = 4                     3   j = 2                   5 1   j = 0                 7 3     j = -2               11 5       j = -4             9 8         j = -6           11 10           j = -8         8 10             j = -10       5 10               j = -12     3 8                 j = -14   1 5                   j = -16   3                     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, Alternating, 168]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 168]]
Out[4]=	PD[X[8, 1, 9, 2], X[18, 11, 19, 12], 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[20, 13, 21, 14], > X[12, 19, 13, 20], X[22, 16, 7, 15], X[4, 22, 5, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -11, 5, -6}, > {6, -1, 7, -3, 2, -9, 8, -5, 10, -7, 4, -2, 9, -8, 11, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 4 8 13 18 20 19 18 12 q - ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q 3/2 5/2 > 8 Sqrt[q] + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 2 2 3 -16 2 2 4 7 4 2 2 -2 - q + --- - --- + --- - q - --- + --- - --- + -- + -- + -- + 2 q - 22 20 18 14 12 10 8 4 2 q q q q q q q q q 4 6 > 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 168]][a, z]
Out[8]=	3 a a 3 3 3 5 3 5 3 5 5 5 -(-) + -- - a z - 4 a z + 8 a z - 4 a z - 4 a z + 12 a z - 4 a z - z z 7 3 7 5 7 3 9 > a z + 6 a z - a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 168]][a, z]
Out[9]=	3 3 2 a a 3 2 2 2 6 2 8 2 2 z -a + - + -- - a z - a z + 2 z + 4 a z - 4 a z - 2 a z - ---- + z z a 3 3 3 5 3 7 3 9 3 4 2 4 > 8 a z + 14 a z - 2 a z - 4 a z + 2 a z - 13 z - 23 a z - 5 4 4 8 4 10 4 3 z 5 3 5 5 5 > 18 a z + 7 a z - a z + ---- - 22 a z - 38 a z + 4 a z + a 7 5 9 5 6 2 6 4 6 6 6 8 6 > 13 a z - 4 a z + 14 z + 22 a z + 30 a z + 14 a z - 8 a z - 7 z 7 3 7 5 7 7 7 8 4 8 6 8 > -- + 21 a z + 42 a z + 9 a z - 11 a z - 4 z - 7 a z - 11 a z - a 9 3 9 5 9 2 10 4 10 > 6 a z - 14 a z - 8 a z - 3 a z - 3 a z
In[10]:=	Kh[L][q, t]
Out[10]=	8 11 1 3 1 5 3 8 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 18 7 16 6 14 6 14 5 12 5 12 4 10 4 q q q t q t q t q t q t q t q t 10 8 10 11 10 9 5 t 2 2 2 > ------ + ----- + ----- + ----- + ---- + ---- + 7 t + --- + 3 t + 5 q t + 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q 2 3 4 3 6 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a168

L11a167 L11a169