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L11n441 L11n443

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- 3q-3/2 + 5q-1/2 - 10q1/2 + 10q3/2 - 14q5/2 + 10q7/2 - 10q9/2 + 6q11/2 - 3q13/2 + q15/2

A2 (sl(3)) Invariant:

3q-6 + 3q-4 + 2q-2 + 9 + 7q2 + 11q4 + 13q6 + 10q8 + 12q10 + 5q12 + 6q14 + 3q16 - 2q18 + q20 - q22 - q24

HOMFLY-PT Polynomial:

a-7z-1 + a-7z - a-5z-3 - 5a-5z-1 - 6a-5z - 3a-5z3 + 3a-3z-3 + 10a-3z-1 + 13a-3z + 7a-3z3 + 2a-3z5 - 3a-1z-3 - 9a-1z-1 - 11a-1z - 5a-1z3 + az-3 + 3az-1 + 3az

Kauffman Polynomial:

a-8 - 3a-8z2 + 3a-8z4 - a-8z6 - 2a-7z-1 + 6a-7z - 9a-7z3 + 9a-7z5 - 3a-7z7 + 6a-6 - 14a-6z2 + 11a-6z4 + 3a-6z6 - 3a-6z8 + a-5z-3 - 11a-5z-1 + 30a-5z - 47a-5z3 + 40a-5z5 - 10a-5z7 - a-5z9 - 3a-4z-2 + 18a-4 - 28a-4z2 + 9a-4z4 + 14a-4z6 - 8a-4z8 + 3a-3z-3 - 18a-3z-1 + 49a-3z - 74a-3z3 + 53a-3z5 - 14a-3z7 - a-3z9 - 6a-2z-2 + 21a-2 - 24a-2z2 + 4a-2z4 + 7a-2z6 - 5a-2z8 + 3a-1z-3 - 14a-1z-1 + 35a-1z - 42a-1z3 + 22a-1z5 - 7a-1z7 - 3z-2 + 9 - 7z2 + 3z4 - 3z6 + az-3 - 5az-1 + 10az - 6az3

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 16                   1 j = 14                 2   j = 12               4 1   j = 10             6 2     j = 8           5 5       j = 6         9 5         j = 4       4 8           j = 2     6 6             j = 0   2 7               j = -2 1 3                 j = -4 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]=	4
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 442]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 442]]
Out[4]=	PD[X[6, 1, 7, 2], X[2, 5, 3, 6], X[11, 19, 12, 18], X[3, 11, 4, 10], > X[9, 1, 10, 4], X[7, 17, 8, 16], X[15, 5, 16, 8], X[13, 20, 14, 21], > X[19, 15, 20, 22], X[21, 12, 22, 13], X[17, 9, 18, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, -3, 10, -8, 11}, > {-7, 6, -11, 3, -9, 8, -10, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-3 5 3/2 5/2 7/2 9/2 11/2 ---- + ------- - 10 Sqrt[q] + 10 q - 14 q + 10 q - 10 q + 6 q - 3/2 Sqrt[q] q 13/2 15/2 > 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	3 3 2 2 4 6 8 10 12 14 9 + -- + -- + -- + 7 q + 11 q + 13 q + 10 q + 12 q + 5 q + 6 q + 6 4 2 q q q 16 18 20 22 24 > 3 q - 2 q + q - q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 442]][a, z]
Out[8]=	1 3 3 a 1 5 10 9 3 a z 6 z -(-----) + ----- - ---- + -- + ---- - ---- + ---- - --- + --- + -- - --- + 5 3 3 3 3 3 7 5 3 a z z 7 5 a z a z a z z a z a z a z a a 3 3 3 5 13 z 11 z 3 z 7 z 5 z 2 z > ---- - ---- + 3 a z - ---- + ---- - ---- + ---- 3 a 5 3 a 3 a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 442]][a, z]
Out[9]=	-8 6 18 21 1 3 3 a 3 3 6 9 + a + -- + -- + -- + ----- + ----- + ---- + -- - -- - ----- - ----- - 6 4 2 5 3 3 3 3 3 2 4 2 2 2 a a a a z a z a z z z a z a z 2 11 18 14 5 a 6 z 30 z 49 z 35 z 2 > ---- - ---- - ---- - --- - --- + --- + ---- + ---- + ---- + 10 a z - 7 z - 7 5 3 a z z 7 5 3 a a z a z a z a a a 2 2 2 2 3 3 3 3 3 z 14 z 28 z 24 z 9 z 47 z 74 z 42 z 3 > ---- - ----- - ----- - ----- - ---- - ----- - ----- - ----- - 6 a z + 8 6 4 2 7 5 3 a a a a a a a a 4 4 4 4 5 5 5 5 4 3 z 11 z 9 z 4 z 9 z 40 z 53 z 22 z 6 > 3 z + ---- + ----- + ---- + ---- + ---- + ----- + ----- + ----- - 3 z - 8 6 4 2 7 5 3 a a a a a a a a 6 6 6 6 7 7 7 7 8 8 z 3 z 14 z 7 z 3 z 10 z 14 z 7 z 3 z 8 z > -- + ---- + ----- + ---- - ---- - ----- - ----- - ---- - ---- - ---- - 8 6 4 2 7 5 3 a 6 4 a a a a a a a a a 8 9 9 5 z z z > ---- - -- - -- 2 5 3 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 3 1 2 3 2 4 4 2 6 2 7 + 6 q + ----- + ----- + - + ---- + 6 q t + 4 q t + 8 q t + 9 q t + 4 2 2 2 t 2 q t q t q t 6 3 8 3 8 4 10 4 10 5 12 5 12 6 > 5 q t + 5 q t + 5 q t + 6 q t + 2 q t + 4 q t + q t + 14 6 16 7 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n442

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