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L11a444 L11a446

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DrawMorseLink

PD Presentation:

X6172 X14,4,15,3 X18,10,19,9 X16,8,17,7 X20,11,21,12 X22,20,13,19 X10,21,11,22 X8,14,9,13 X12,18,5,17 X2536 X4,16,1,15

Gauss Code:

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

Jones Polynomial:

q-3 - 3q-2 + 7q-1 - 11 + 17q - 19q2 + 21q3 - 17q4 + 14q5 - 9q6 + 4q7 - q8

A2 (sl(3)) Invariant:

q-10 - q-6 + 3q-4 - q-2 + 1 + 6q2 + 7q6 + 3q8 + 4q10 + 5q12 - 2q14 + 4q16 - 2q18 - 2q20 + 2q22 - q24

HOMFLY-PT Polynomial:

- a-6 - a-6z2 - a-6z4 + a-4z-2 + 3a-4 + 3a-4z2 + 2a-4z4 + a-4z6 - 2a-2z-2 - 3a-2 - a-2z2 + a-2z4 + a-2z6 + z-2 - 3z2 - 2z4 + a2 + a2z2

Kauffman Polynomial:

- a-9z3 + a-9z5 + a-8z2 - 5a-8z4 + 4a-8z6 - 2a-7z + 6a-7z3 - 13a-7z5 + 8a-7z7 + 3a-6 - 7a-6z2 + 11a-6z4 - 15a-6z6 + 9a-6z8 - 7a-5z + 20a-5z3 - 20a-5z5 + 3a-5z7 + 5a-5z9 - a-4z-2 + 11a-4 - 31a-4z2 + 47a-4z4 - 42a-4z6 + 16a-4z8 + a-4z10 + 2a-3z-1 - 12a-3z + 24a-3z3 - 15a-3z5 - 6a-3z7 + 8a-3z9 - 2a-2z-2 + 11a-2 - 26a-2z2 + 35a-2z4 - 31a-2z6 + 11a-2z8 + a-2z10 + 2a-1z-1 - 8a-1z + 17a-1z3 - 17a-1z5 + 2a-1z7 + 3a-1z9 - z-2 + 3 + z4 - 7z6 + 4z8 - az + 6az3 - 8az5 + 3az7 - a2 + 3a2z2 - 3a2z4 + a2z6

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 17                       1 j = 15                     3   j = 13                   6 1   j = 11                 8 3     j = 9               10 7       j = 7             11 7         j = 5           8 10           j = 3         9 11             j = 1       5 11               j = -1     2 6                 j = -3   1 5                   j = -5   2                     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, Alternating, 445]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 445]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 4, 15, 3], X[18, 10, 19, 9], X[16, 8, 17, 7], > X[20, 11, 21, 12], X[22, 20, 13, 19], X[10, 21, 11, 22], X[8, 14, 9, 13], > X[12, 18, 5, 17], X[2, 5, 3, 6], X[4, 16, 1, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, -8, 3, -7, 5, -9}, > {8, -2, 11, -4, 9, -3, 6, -5, 7, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-3 3 7 2 3 4 5 6 7 8 -11 + q - -- + - + 17 q - 19 q + 21 q - 17 q + 14 q - 9 q + 4 q - q 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 -6 3 -2 2 6 8 10 12 14 1 + q - q + -- - q + 6 q + 7 q + 3 q + 4 q + 5 q - 2 q + 4 q 16 18 20 22 24 > 4 q - 2 q - 2 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 445]][a, z]
Out[8]=	2 2 2 -6 3 3 2 -2 1 2 2 z 3 z z 2 2 -a + -- - -- + a + z + ----- - ----- - 3 z - -- + ---- - -- + a z - 4 2 4 2 2 2 6 4 2 a a a z a z a a a 4 4 4 6 6 4 z 2 z z z z > 2 z - -- + ---- + -- + -- + -- 6 4 2 4 2 a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 445]][a, z]
Out[9]=	3 11 11 2 -2 1 2 2 2 2 z 7 z 12 z 3 + -- + -- + -- - a - z - ----- - ----- + ---- + --- - --- - --- - ---- - 6 4 2 4 2 2 2 3 a z 7 5 3 a a a a z a z a z a a a 2 2 2 2 3 3 3 8 z z 7 z 31 z 26 z 2 2 z 6 z 20 z > --- - a z + -- - ---- - ----- - ----- + 3 a z - -- + ---- + ----- + a 8 6 4 2 9 7 5 a a a a a a a 3 3 4 4 4 4 5 24 z 17 z 3 4 5 z 11 z 47 z 35 z 2 4 z > ----- + ----- + 6 a z + z - ---- + ----- + ----- + ----- - 3 a z + -- - 3 a 8 6 4 2 9 a a a a a a 5 5 5 5 6 6 6 13 z 20 z 15 z 17 z 5 6 4 z 15 z 42 z > ----- - ----- - ----- - ----- - 8 a z - 7 z + ---- - ----- - ----- - 7 5 3 a 8 6 4 a a a a a a 6 7 7 7 7 8 8 31 z 2 6 8 z 3 z 6 z 2 z 7 8 9 z 16 z > ----- + a z + ---- + ---- - ---- + ---- + 3 a z + 4 z + ---- + ----- + 2 7 5 3 a 6 4 a a a a a a 8 9 9 9 10 10 11 z 5 z 8 z 3 z z z > ----- + ---- + ---- + ---- + --- + --- 2 5 3 a 4 2 a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 2 1 5 2 6 5 q 3 11 q + 9 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 11 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 8 q t + 10 q t + 11 q t + 7 q t + 10 q t + 7 q t + 8 q t + 11 5 13 5 13 6 15 6 17 7 > 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a445

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