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L11a165 L11a167

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-17/2 - 3q-15/2 + 6q-13/2 - 10q-11/2 + 12q-9/2 - 14q-7/2 + 13q-5/2 - 11q-3/2 + 8q-1/2 - 5q1/2 + 2q3/2 - q5/2

A2 (sl(3)) Invariant:

- q-26 + q-22 - q-20 + 3q-18 + q-14 + 2q-12 - 2q-10 + 3q-8 - 2q-6 + q-4 + q-2 - 1 + 2q2 + q6 + q8

HOMFLY-PT Polynomial:

- a-1z-1 - 3a-1z - a-1z3 + 2az-1 + 6az + 7az3 + 2az5 - 2a3z-1 - 6a3z - 6a3z3 - 4a3z5 - a3z7 + a5z-1 + 4a5z + 6a5z3 + 2a5z5 - 2a7z - a7z3

Kauffman Polynomial:

- a-1z-1 + 5a-1z - 8a-1z3 + 5a-1z5 - a-1z7 + 3z2 - 9z4 + 8z6 - 2z8 - 2az-1 + 11az - 18az3 + 7az5 + 4az7 - 2az9 - a2 + 3a2z2 - 13a2z4 + 14a2z6 - 2a2z8 - a2z10 - 2a3z-1 + 10a3z - 15a3z3 + 4a3z5 + 10a3z7 - 5a3z9 + a4z2 - 8a4z4 + 15a4z6 - 5a4z8 - a4z10 - a5z-1 + 6a5z - 12a5z3 + 12a5z5 - a5z7 - 3a5z9 - 2a6z2 + 2a6z4 + 4a6z6 - 5a6z8 + a7z - 4a7z3 + 7a7z5 - 6a7z7 - 2a8z2 + 5a8z4 - 5a8z6 - a9z + 3a9z3 - 3a9z5 + a10z2 - 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                     1   j = 2                   4 1   j = 0                 4 1     j = -2               7 4       j = -4             7 5         j = -6           7 6           j = -8         6 8             j = -10       4 6               j = -12     2 6                 j = -14   1 4                   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, Alternating, 166]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 166]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[6, 7, 1, 8], > X[16, 13, 17, 14], X[14, 6, 15, 5], X[4, 16, 5, 15], X[20, 11, 21, 12], > X[22, 18, 7, 17], X[18, 22, 19, 21], X[12, 19, 13, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -7, 6, -4}, > {4, -1, 2, -3, 8, -11, 5, -6, 7, -5, 9, -10, 11, -8, 10, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 3 6 10 12 14 13 11 8 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 > 5 Sqrt[q] + 2 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 -22 -20 3 -14 2 2 3 2 -4 -2 2 -1 - q + q - q + --- + q + --- - --- + -- - -- + q + q + 2 q + 18 12 10 8 6 q q q q q 6 8 > q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 166]][a, z]
Out[8]=	3 5 3 1 2 a 2 a a 3 z 3 5 7 z -(---) + --- - ---- + -- - --- + 6 a z - 6 a z + 4 a z - 2 a z - -- + a z z z z a a 3 3 3 5 3 7 3 5 3 5 5 5 3 7 > 7 a z - 6 a z + 6 a z - a z + 2 a z - 4 a z + 2 a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 166]][a, z]
Out[9]=	3 5 2 1 2 a 2 a a 5 z 3 5 7 9 -a - --- - --- - ---- - -- + --- + 11 a z + 10 a z + 6 a z + a z - a z + a z z z z a 3 2 2 2 4 2 6 2 8 2 10 2 8 z 3 > 3 z + 3 a z + a z - 2 a z - 2 a z + a z - ---- - 18 a z - a 3 3 5 3 7 3 9 3 4 2 4 4 4 > 15 a z - 12 a z - 4 a z + 3 a z - 9 z - 13 a z - 8 a z + 5 6 4 8 4 10 4 5 z 5 3 5 5 5 7 5 > 2 a z + 5 a z - a z + ---- + 7 a z + 4 a z + 12 a z + 7 a z - a 7 9 5 6 2 6 4 6 6 6 8 6 z 7 > 3 a z + 8 z + 14 a z + 15 a z + 4 a z - 5 a z - -- + 4 a z + a 3 7 5 7 7 7 8 2 8 4 8 6 8 9 > 10 a z - a z - 6 a z - 2 z - 2 a z - 5 a z - 5 a z - 2 a z - 3 9 5 9 2 10 4 10 > 5 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 7 1 2 1 4 2 6 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 6 6 8 7 6 7 4 t 2 2 2 > ------ + ----- + ----- + ----- + ---- + ---- + 4 t + --- + t + 4 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 + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a166

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