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L11n332 L11n334

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DrawMorseLink

PD Presentation:

X6172 X11,16,12,17 X3849 X2,18,3,17 X5,14,6,15 X18,7,19,8 X15,12,16,5 X13,20,14,21 X9,13,10,22 X21,11,22,10 X19,1,20,4

Gauss Code:

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

Jones Polynomial:

- q-7 + 2q-6 - 3q-5 + 6q-4 - 5q-3 + 6q-2 - 5q-1 + 5 - 2q + q2

A2 (sl(3)) Invariant:

- q-22 - q-16 + 3q-14 + 3q-12 + 4q-10 + 5q-8 + 3q-6 + 4q-4 + 2q-2 + 2 + 2q2 - q4 + q6 + q8

HOMFLY-PT Polynomial:

a-2 + z-2 - 2z2 - 2a2z-2 - 3a2 + a2z4 + a4z-2 + 3a4 + 2a4z2 + a4z4 - a6 - a6z2

Kauffman Polynomial:

- a-2 + a-2z2 - a-1z + 2a-1z3 - z-2 + 3 + z2 - z4 + z6 + 2az-1 - 8az + 15az3 - 11az5 + 3az7 - 2a2z-2 + 11a2 - 20a2z2 + 19a2z4 - 12a2z6 + 3a2z8 + 2a3z-1 - 12a3z + 23a3z3 - 16a3z5 + a3z7 + a3z9 - a4z-2 + 11a4 - 31a4z2 + 36a4z4 - 23a4z6 + 5a4z8 - 7a5z + 17a5z3 - 10a5z5 - a5z7 + a5z9 + 3a6 - 11a6z2 + 16a6z4 - 10a6z6 + 2a6z8 - 2a7z + 7a7z3 - 5a7z5 + a7z7

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 5                   1 j = 3                 1   j = 1               4 1   j = -1             4 4     j = -3           2 2 1     j = -5         3 4         j = -7       3 2           j = -9     1 4             j = -11   1 2               j = -13   1                 j = -15 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 333]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 333]]
Out[4]=	PD[X[6, 1, 7, 2], X[11, 16, 12, 17], X[3, 8, 4, 9], X[2, 18, 3, 17], > X[5, 14, 6, 15], X[18, 7, 19, 8], X[15, 12, 16, 5], X[13, 20, 14, 21], > X[9, 13, 10, 22], X[21, 11, 22, 10], X[19, 1, 20, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, -3, 11}, {-5, -1, 6, 3, -9, 10, -2, 7}, > {-8, 5, -7, 2, 4, -6, -11, 8, -10, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-7 2 3 6 5 6 5 2 5 - q + -- - -- + -- - -- + -- - - - 2 q + q 6 5 4 3 2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -16 3 3 4 5 3 4 2 2 4 6 8 2 - q - q + --- + --- + --- + -- + -- + -- + -- + 2 q - q + q + q 14 12 10 8 6 4 2 q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 333]][a, z]
Out[8]=	2 4 -2 2 4 6 -2 2 a a 2 4 2 6 2 2 4 a - 3 a + 3 a - a + z - ---- + -- - 2 z + 2 a z - a z + a z + 2 2 z z 4 4 > a z
In[9]:=	Kauffman[Link[11, NonAlternating, 333]][a, z]
Out[9]=	2 4 3 -2 2 4 6 -2 2 a a 2 a 2 a z 3 - a + 11 a + 11 a + 3 a - z - ---- - -- + --- + ---- - - - 8 a z - 2 2 z z a z z 2 3 5 7 2 z 2 2 4 2 6 2 > 12 a z - 7 a z - 2 a z + z + -- - 20 a z - 31 a z - 11 a z + 2 a 3 2 z 3 3 3 5 3 7 3 4 2 4 4 4 > ---- + 15 a z + 23 a z + 17 a z + 7 a z - z + 19 a z + 36 a z + a 6 4 5 3 5 5 5 7 5 6 2 6 > 16 a z - 11 a z - 16 a z - 10 a z - 5 a z + z - 12 a z - 4 6 6 6 7 3 7 5 7 7 7 2 8 4 8 > 23 a z - 10 a z + 3 a z + a z - a z + a z + 3 a z + 5 a z + 6 8 3 9 5 9 > 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-3 4 1 1 1 2 1 4 3 q + - + 4 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + q 15 7 13 6 11 6 11 5 9 5 9 4 7 4 q t q t q t q t q t q t q t 2 3 4 2 2 4 3 5 2 > ----- + ----- + ----- + ----- + ---- + --- + q t + q t + q t 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n333

L11n332 L11n334