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L11n433 L11n435

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DrawMorseLink

PD Presentation:

X8192 X14,4,15,3 X12,14,7,13 X2738 X22,10,13,9 X6,22,1,21 X15,20,16,21 X5,17,6,16 X18,11,19,12 X10,17,11,18 X19,5,20,4

Gauss Code:

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

Jones Polynomial:

- q-3 + 4q-2 - 6q-1 + 9 - 9q + 10q2 - 7q3 + 6q4 - 3q5 + q6

A2 (sl(3)) Invariant:

- q-10 + q-8 + 2q-6 - q-4 + 3q-2 + 1 + 3q2 + 5q4 + 3q6 + 6q8 + q10 + 2q12 + 2q14 - q16 + q18

HOMFLY-PT Polynomial:

a-4z-2 + 2a-4 + 2a-4z2 + a-4z4 - 2a-2z-2 - 5a-2 - 7a-2z2 - 4a-2z4 - a-2z6 + z-2 + 3 + 4z2 + 2z4 - a2z2

Kauffman Polynomial:

2a-6z2 - 3a-6z4 + a-6z6 + 5a-5z3 - 9a-5z5 + 3a-5z7 - a-4z-2 + 6a-4 - 11a-4z2 + 13a-4z4 - 13a-4z6 + 4a-4z8 + 2a-3z-1 - 6a-3z + 8a-3z3 - 7a-3z5 - 2a-3z7 + 2a-3z9 - 2a-2z-2 + 12a-2 - 33a-2z2 + 44a-2z4 - 30a-2z6 + 8a-2z8 + 2a-1z-1 - 8a-1z + 10a-1z3 - 2a-1z5 - 3a-1z7 + 2a-1z9 - z-2 + 8 - 24z2 + 32z4 - 16z6 + 4z8 - 3az + 8az3 - 4az5 + 2az7 + a2 - 4a2z2 + 4a2z4 - a3z + a3z3

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                   1 j = 11                 2   j = 9               4 1   j = 7             4 3     j = 5           6 3       j = 3         4 5         j = 1       5 5           j = -1     2 5             j = -3   2 4               j = -5   3                 j = -7 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, 434]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 434]]
Out[4]=	PD[X[8, 1, 9, 2], X[14, 4, 15, 3], X[12, 14, 7, 13], X[2, 7, 3, 8], > X[22, 10, 13, 9], X[6, 22, 1, 21], X[15, 20, 16, 21], X[5, 17, 6, 16], > X[18, 11, 19, 12], X[10, 17, 11, 18], X[19, 5, 20, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, 11, -8, -6}, {4, -1, 5, -10, 9, -3}, > {3, -2, -7, 8, 10, -9, -11, 7, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-3 4 6 2 3 4 5 6 9 - q + -- - - - 9 q + 10 q - 7 q + 6 q - 3 q + q 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 -8 2 -4 3 2 4 6 8 10 12 1 - q + q + -- - q + -- + 3 q + 5 q + 3 q + 6 q + q + 2 q + 6 2 q q 14 16 18 > 2 q - q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 434]][a, z]
Out[8]=	2 2 4 2 5 -2 1 2 2 2 z 7 z 2 2 4 z 3 + -- - -- + z + ----- - ----- + 4 z + ---- - ---- - a z + 2 z + -- - 4 2 4 2 2 2 4 2 4 a a a z a z a a a 4 6 4 z z > ---- - -- 2 2 a a
In[9]:=	Kauffman[Link[11, NonAlternating, 434]][a, z]
Out[9]=	6 12 2 -2 1 2 2 2 6 z 8 z 8 + -- + -- + a - z - ----- - ----- + ---- + --- - --- - --- - 3 a z - 4 2 4 2 2 2 3 a z 3 a a a a z a z a z a 2 2 2 3 3 3 3 2 2 z 11 z 33 z 2 2 5 z 8 z 10 z > a z - 24 z + ---- - ----- - ----- - 4 a z + ---- + ---- + ----- + 6 4 2 5 3 a a a a a a 4 4 4 5 5 3 3 3 4 3 z 13 z 44 z 2 4 9 z 7 z > 8 a z + a z + 32 z - ---- + ----- + ----- + 4 a z - ---- - ---- - 6 4 2 5 3 a a a a a 5 6 6 6 7 7 7 2 z 5 6 z 13 z 30 z 3 z 2 z 3 z 7 > ---- - 4 a z - 16 z + -- - ----- - ----- + ---- - ---- - ---- + 2 a z + a 6 4 2 5 3 a a a a a a 8 8 9 9 8 4 z 8 z 2 z 2 z > 4 z + ---- + ---- + ---- + ---- 4 2 3 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	5 1 3 2 4 2 3 3 2 - + 5 q + ----- + ----- + ----- + ---- + --- + 5 q t + 4 q t + 5 q t + q 7 3 5 2 3 2 3 q t q t q t q t q t 5 2 5 3 7 3 7 4 9 4 9 5 11 5 13 6 > 6 q t + 3 q t + 4 q t + 3 q t + 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n434

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