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L11n40 L11n42

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DrawMorseLink

PD Presentation:

X6172 X12,7,13,8 X4,13,1,14 X9,18,10,19 X8493 X5,14,6,15 X15,20,16,21 X17,22,18,5 X21,16,22,17 X19,10,20,11 X2,12,3,11

Gauss Code:

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

Jones Polynomial:

q-19/2 - 2q-17/2 + 5q-15/2 - 6q-13/2 + 6q-11/2 - 7q-9/2 + 5q-7/2 - 5q-5/2 + 2q-3/2 - q-1/2

A2 (sl(3)) Invariant:

- 2q-28 - q-26 - 3q-24 - q-22 + 4q-16 + q-14 + 4q-12 + 2q-10 + 2q-8 + 2q-6 + q-2

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- 2a3z-1 + 7a3z - 9a3z3 + 5a3z5 - a3z7 + 3a4 - 2a4z2 - 7a4z4 + 8a4z6 - 2a4z8 - 2a5z-1 + 12a5z - 26a5z3 + 18a5z5 - a5z7 - a5z9 + 2a6z2 - 14a6z4 + 18a6z6 - 5a6z8 + a7z-1 + 8a7z - 27a7z3 + 23a7z5 - 3a7z7 - a7z9 - 3a8 + 3a8z2 - 6a8z4 + 9a8z6 - 3a8z8 + a9z-1 + 4a9z - 12a9z3 + 10a9z5 - 3a9z7 - 2a10z2 + a10z4 - a10z6 + a11z - 2a11z3 + a12 - a12z2

Khovanov Homology:

trqj 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                 1   j = -4               4 1   j = -6             3 3     j = -8           4 2       j = -10       1 3 3         j = -12       4 4           j = -14     2 3             j = -16   1 4               j = -18   1                 j = -20 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, 41]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 41]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 7, 13, 8], X[4, 13, 1, 14], X[9, 18, 10, 19], > X[8, 4, 9, 3], X[5, 14, 6, 15], X[15, 20, 16, 21], X[17, 22, 18, 5], > X[21, 16, 22, 17], X[19, 10, 20, 11], X[2, 12, 3, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {-6, -1, 2, -5, -4, 10, 11, -2, 3, 6, -7, 9, -8, 4, > -10, 7, -9, 8}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 2 5 6 6 7 5 5 2 1 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]=	-2 -26 3 -22 4 -14 4 2 2 2 -2 --- - q - --- - q + --- + q + --- + --- + -- + -- + q 28 24 16 12 10 8 6 q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 41]][a, z]
Out[8]=	3 5 7 9 -2 a 2 a a a 3 5 7 3 3 5 3 7 3 ----- + ---- + -- - -- - 5 a z + 6 a z - a z - 4 a z + 8 a z - 3 a z - z z z z 3 5 5 5 7 5 5 7 > a z + 5 a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 41]][a, z]
Out[9]=	3 5 7 9 4 8 12 2 a 2 a a a 3 5 7 3 a - 3 a + a - ---- - ---- + -- + -- + 7 a z + 12 a z + 8 a z + z z z z 9 11 4 2 6 2 8 2 10 2 12 2 > 4 a z + a z - 2 a z + 2 a z + 3 a z - 2 a z - a z - 3 3 5 3 7 3 9 3 11 3 4 4 6 4 > 9 a z - 26 a z - 27 a z - 12 a z - 2 a z - 7 a z - 14 a z - 8 4 10 4 3 5 5 5 7 5 9 5 4 6 > 6 a z + a z + 5 a z + 18 a z + 23 a z + 10 a z + 8 a z + 6 6 8 6 10 6 3 7 5 7 7 7 9 7 4 8 > 18 a z + 9 a z - a z - a z - a z - 3 a z - 3 a z - 2 a z - 6 8 8 8 5 9 7 9 > 5 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 4 1 1 1 4 2 3 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 6 4 20 7 18 6 16 6 16 5 14 5 14 4 12 4 q q q t q t q t q t q t q t q t 1 4 3 3 4 2 3 t t 2 > ------ + ------ + ------ + ------ + ----- + ---- + ---- + -- + -- + t 10 4 12 3 10 3 10 2 8 2 8 6 4 2 q t q t q t q t q t q t q t q q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n41

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