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L11n101 L11n103

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DrawMorseLink

PD Presentation:

X6172 X3,13,4,12 X7,16,8,17 X17,22,18,5 X13,18,14,19 X21,14,22,15 X9,20,10,21 X15,8,16,9 X19,10,20,11 X2536 X11,1,12,4

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

- q-28 - 4q-26 - 2q-22 - 2q-20 + 4q-18 + 5q-14 + 2q-12 + 2q-10 + 4q-8 - 2q-6 + 3q-4 - 1 + q2

HOMFLY-PT Polynomial:

- az - az3 - 2a3z-1 - 3a3z + a3z5 + 2a5z-1 + 5a5z + 5a5z3 + 2a5z5 + a7z-1 - a7z - 2a7z3 - a9z-1

Kauffman Polynomial:

- az + 2az3 - az5 - a2z2 + 5a2z4 - 3a2z6 - 2a3z-1 + 7a3z - 7a3z3 + 9a3z5 - 5a3z7 + 2a4 - 3a4z2 - a4z4 + 7a4z6 - 5a4z8 - 2a5z-1 + 15a5z - 33a5z3 + 27a5z5 - 7a5z7 - 2a5z9 - 4a6 + 12a6z2 - 24a6z4 + 23a6z6 - 10a6z8 + a7z-1 + 9a7z - 25a7z3 + 20a7z5 - 5a7z7 - 2a7z9 - 9a8 + 25a8z2 - 24a8z4 + 13a8z6 - 5a8z8 + a9z-1 + 2a9z - a9z3 + 3a9z5 - 3a9z7 - 4a10 + 11a10z2 - 6a10z4

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 2                   1 j = 0                 2   j = -2               5 1   j = -4             7 4     j = -6           7 3       j = -8         6 7         j = -10       7 7           j = -12     3 6             j = -14   3 7               j = -16   3                 j = -18 3

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, 102]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 102]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 13, 4, 12], X[7, 16, 8, 17], X[17, 22, 18, 5], > X[13, 18, 14, 19], X[21, 14, 22, 15], X[9, 20, 10, 21], X[15, 8, 16, 9], > X[19, 10, 20, 11], X[2, 5, 3, 6], X[11, 1, 12, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -2, 11}, {10, -1, -3, 8, -7, 9, -11, 2, -5, 6, -8, 3, -4, 5, > -9, 7, -6, 4}]
In[6]:=	Jones[L][q]
Out[6]=	3 6 10 13 13 14 10 7 3 ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- - Sqrt[q] 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 4 2 2 4 5 2 2 4 2 3 2 -1 - q - --- - --- - --- + --- + --- + --- + --- + -- - -- + -- + q 26 22 20 18 14 12 10 8 6 4 q q q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 102]][a, z]
Out[8]=	3 5 7 9 -2 a 2 a a a 3 5 7 3 5 3 ----- + ---- + -- - -- - a z - 3 a z + 5 a z - a z - a z + 5 a z - z z z z 7 3 3 5 5 5 > 2 a z + a z + 2 a z
In[9]:=	Kauffman[Link[11, NonAlternating, 102]][a, z]
Out[9]=	3 5 7 9 4 6 8 10 2 a 2 a a a 3 5 2 a - 4 a - 9 a - 4 a - ---- - ---- + -- + -- - a z + 7 a z + 15 a z + z z z z 7 9 2 2 4 2 6 2 8 2 10 2 > 9 a z + 2 a z - a z - 3 a z + 12 a z + 25 a z + 11 a z + 3 3 3 5 3 7 3 9 3 2 4 4 4 > 2 a z - 7 a z - 33 a z - 25 a z - a z + 5 a z - a z - 6 4 8 4 10 4 5 3 5 5 5 7 5 > 24 a z - 24 a z - 6 a z - a z + 9 a z + 27 a z + 20 a z + 9 5 2 6 4 6 6 6 8 6 3 7 5 7 > 3 a z - 3 a z + 7 a z + 23 a z + 13 a z - 5 a z - 7 a z - 7 7 9 7 4 8 6 8 8 8 5 9 7 9 > 5 a z - 3 a z - 5 a z - 10 a z - 5 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 5 3 3 3 7 3 6 7 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 7 6 7 7 3 7 t 2 2 > ------ + ----- + ----- + ----- + ---- + ---- + 2 t + -- + 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

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n102

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