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L11n105 L11n107

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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X7,16,8,17 X17,22,18,5 X13,18,14,19 X9,21,10,20 X19,14,20,15 X21,9,22,8 X15,10,16,11 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

q-21/2 - 2q-19/2 + 3q-17/2 - 3q-15/2 + 2q-13/2 - 3q-11/2 + q-9/2 - q-5/2 + q-3/2 - q-1/2

A2 (sl(3)) Invariant:

- q-34 - 2q-32 + q-30 - q-28 + 2q-24 + q-22 + 3q-20 + q-18 + 2q-16 - q-12 + q-10 + q-8 + q-4 + q-2

HOMFLY-PT Polynomial:

- a3z-1 - 5a3z - 5a3z3 - a3z5 + 2a5z-1 + 8a5z + 10a5z3 + 6a5z5 + a5z7 - 3a7z-1 - 9a7z - 7a7z3 - a7z5 + 3a9z-1 + 4a9z - a11z-1

Kauffman Polynomial:

- a3z-1 + 6a3z - 10a3z3 + 6a3z5 - a3z7 + 5a4z2 - 9a4z4 + 6a4z6 - a4z8 - 2a5z-1 + 12a5z - 23a5z3 + 14a5z5 - 2a5z7 - 2a6 + 14a6z2 - 25a6z4 + 14a6z6 - 2a6z8 - 3a7z-1 + 14a7z - 22a7z3 + 5a7z5 + 4a7z7 - a7z9 + 9a8z2 - 24a8z4 + 17a8z6 - 3a8z8 - 3a9z-1 + 11a9z - 15a9z3 + 5a9z5 + 3a9z7 - a9z9 + 2a10 - 3a10z2 - 4a10z4 + 8a10z6 - 2a10z8 - a11z-1 + 3a11z - 6a11z3 + 8a11z5 - 2a11z7 + a12 - 3a12z2 + 4a12z4 - a12z6

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 0                       1 j = -2                         j = -4                 1 2 1   j = -6               1 1 1     j = -8             2 2 1       j = -10           3 2 1         j = -12         2 4 1           j = -14       2 2 1             j = -16     1 2 1               j = -18   1 2                   j = -20   1                     j = -22 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, 106]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 106]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[7, 16, 8, 17], X[17, 22, 18, 5], > X[13, 18, 14, 19], X[9, 21, 10, 20], X[19, 14, 20, 15], X[21, 9, 22, 8], > X[15, 10, 16, 11], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -3, 8, -6, 9, 11, -2, -5, 7, -9, 3, -4, 5, > -7, 6, -8, 4}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 2 3 3 2 3 -(9/2) -(5/2) q - ----- + ----- - ----- + ----- - ----- + q - q + 19/2 17/2 15/2 13/2 11/2 q q q q q -(3/2) 1 > q - ------- Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 2 -30 -28 2 -22 3 -18 2 -12 -10 -8 -q - --- + q - q + --- + q + --- + q + --- - q + q + q + 32 24 20 16 q q q q -4 -2 > q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 106]][a, z]
Out[8]=	3 5 7 9 11 a 2 a 3 a 3 a a 3 5 7 9 -(--) + ---- - ---- + ---- - --- - 5 a z + 8 a z - 9 a z + 4 a z - z z z z z 3 3 5 3 7 3 3 5 5 5 7 5 5 7 > 5 a z + 10 a z - 7 a z - a z + 6 a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 106]][a, z]
Out[9]=	3 5 7 9 11 6 10 12 a 2 a 3 a 3 a a 3 5 -2 a + 2 a + a - -- - ---- - ---- - ---- - --- + 6 a z + 12 a z + z z z z z 7 9 11 4 2 6 2 8 2 10 2 > 14 a z + 11 a z + 3 a z + 5 a z + 14 a z + 9 a z - 3 a z - 12 2 3 3 5 3 7 3 9 3 11 3 4 4 > 3 a z - 10 a z - 23 a z - 22 a z - 15 a z - 6 a z - 9 a z - 6 4 8 4 10 4 12 4 3 5 5 5 7 5 > 25 a z - 24 a z - 4 a z + 4 a z + 6 a z + 14 a z + 5 a z + 9 5 11 5 4 6 6 6 8 6 10 6 12 6 > 5 a z + 8 a z + 6 a z + 14 a z + 17 a z + 8 a z - a z - 3 7 5 7 7 7 9 7 11 7 4 8 6 8 > a z - 2 a z + 4 a z + 3 a z - 2 a z - a z - 2 a z - 8 8 10 8 7 9 9 9 > 3 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 2 1 1 1 2 1 2 2 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 22 9 20 8 18 8 18 7 16 7 16 6 14 6 q q t q t q t q t q t q t q t 1 2 2 1 4 3 1 2 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 16 5 14 5 12 5 14 4 12 4 10 4 12 3 10 3 q t q t q t q t q t q t q t q t 2 1 2 1 1 1 1 t 2 > ----- + ------ + ----- + ----- + ---- + ---- + ---- + -- + t 8 3 10 2 8 2 6 2 8 6 4 4 q t q t q t q t q t q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n106

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