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L11n331 L11n333

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DrawMorseLink

PD Presentation:

X6172 X11,16,12,17 X3849 X17,2,18,3 X5,14,6,15 X18,7,19,8 X15,12,16,5 X13,20,14,21 X9,13,10,22 X21,11,22,10 X4,19,1,20

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-9 + 2q-8 - 3q-7 + 7q-6 - 5q-5 + 6q-4 - 5q-3 + 4q-2 - 2q-1 + 1

A2 (sl(3)) Invariant:

- q-32 + 2q-24 + 2q-22 + 6q-20 + 5q-18 + 5q-16 + 4q-14 + 2q-10 + q-6 + 1

HOMFLY-PT Polynomial:

a2 + 3a2z2 + a2z4 + a4z-2 + a4 - 3a4z4 - a4z6 - 2a6z-2 - 3a6 - 4a6z2 - 4a6z4 - a6z6 + a8z-2 + 2a8 + 4a8z2 + a8z4 - a10

Kauffman Polynomial:

- a2 + 4a2z2 - 4a2z4 + a2z6 - a3z + 5a3z3 - 7a3z5 + 2a3z7 - a4z-2 + 3a4 - 3a4z2 + 3a4z4 - 6a4z6 + 2a4z8 + 2a5z-1 - 8a5z + 10a5z3 - 6a5z5 - a5z7 + a5z9 - 2a6z-2 + 11a6 - 22a6z2 + 27a6z4 - 17a6z6 + 4a6z8 + 2a7z-1 - 12a7z + 14a7z3 - 3a7z5 - 2a7z7 + a7z9 - a8z-2 + 11a8 - 19a8z2 + 22a8z4 - 10a8z6 + 2a8z8 - 7a9z + 10a9z3 - 4a9z5 + a9z7 + 3a10 - 4a10z2 + 2a10z4 - 2a11z + a11z3

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 1                   1 j = -1                 1   j = -3               3 1   j = -5             3 2     j = -7           3 2       j = -9       1 3 3         j = -11       5 3           j = -13     1 5             j = -15   1 2               j = -17   1                 j = -19 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, 332]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 332]]
Out[4]=	PD[X[6, 1, 7, 2], X[11, 16, 12, 17], X[3, 8, 4, 9], X[17, 2, 18, 3], > 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[4, 19, 1, 20]]
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]=	-9 2 3 7 5 6 5 4 2 1 - q + -- - -- + -- - -- + -- - -- + -- - - 8 7 6 5 4 3 2 q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 2 2 6 5 5 4 2 -6 1 - q + --- + --- + --- + --- + --- + --- + --- + q 24 22 20 18 16 14 10 q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 332]][a, z]
Out[8]=	4 6 8 2 4 6 8 10 a 2 a a 2 2 6 2 8 2 a + a - 3 a + 2 a - a + -- - ---- + -- + 3 a z - 4 a z + 4 a z + 2 2 2 z z z 2 4 4 4 6 4 8 4 4 6 6 6 > a z - 3 a z - 4 a z + a z - a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 332]][a, z]
Out[9]=	4 6 8 5 7 2 4 6 8 10 a 2 a a 2 a 2 a 3 -a + 3 a + 11 a + 11 a + 3 a - -- - ---- - -- + ---- + ---- - a z - 2 2 2 z z z z z 5 7 9 11 2 2 4 2 6 2 > 8 a z - 12 a z - 7 a z - 2 a z + 4 a z - 3 a z - 22 a z - 8 2 10 2 3 3 5 3 7 3 9 3 11 3 > 19 a z - 4 a z + 5 a z + 10 a z + 14 a z + 10 a z + a z - 2 4 4 4 6 4 8 4 10 4 3 5 5 5 > 4 a z + 3 a z + 27 a z + 22 a z + 2 a z - 7 a z - 6 a z - 7 5 9 5 2 6 4 6 6 6 8 6 3 7 > 3 a z - 4 a z + a z - 6 a z - 17 a z - 10 a z + 2 a z - 5 7 7 7 9 7 4 8 6 8 8 8 5 9 7 9 > a z - 2 a z + a z + 2 a z + 4 a z + 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 3 1 1 1 2 1 5 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 19 7 17 6 15 6 15 5 13 5 13 4 11 4 q q q t q t q t q t q t q t q t 1 3 3 3 3 2 3 t t 2 > ----- + ------ + ----- + ----- + ----- + ---- + ---- + -- + - + q t 9 4 11 3 9 3 9 2 7 2 7 5 3 q 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 L11n332

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