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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-2 - 2q-1 + 5 - 6q + 9q2 - 9q3 + 11q4 - 9q5 + 7q6 - 5q7 + 3q8 - q9

A2 (sl(3)) Invariant:

q-6 + q-4 + q-2 + 3 + 2q2 + 4q4 + 3q6 + 3q8 + 5q10 + q12 + 4q14 - q22 + q24 - q26

HOMFLY-PT Polynomial:

- 2a-6 - 4a-6z2 - 4a-6z4 - a-6z6 + a-4z-2 + 7a-4 + 14a-4z2 + 13a-4z4 + 6a-4z6 + a-4z8 - 2a-2z-2 - 9a-2 - 15a-2z2 - 10a-2z4 - 2a-2z6 + z-2 + 4 + 4z2 + z4

Kauffman Polynomial:

a-11z3 - a-10z2 + 3a-10z4 + a-9z - 4a-9z3 + 5a-9z5 - a-8 + 2a-8z2 - 7a-8z4 + 6a-8z6 + a-7z - 3a-7z3 - 8a-7z5 + 6a-7z7 + 5a-6z4 - 14a-6z6 + 6a-6z8 - 3a-5z + 9a-5z3 - 3a-5z5 - 9a-5z7 + 4a-5z9 - a-4z-2 + 9a-4 - 31a-4z2 + 56a-4z4 - 39a-4z6 + 6a-4z8 + a-4z10 + 2a-3z-1 - 8a-3z + 4a-3z3 + 24a-3z5 - 25a-3z7 + 6a-3z9 - 2a-2z-2 + 13a-2 - 41a-2z2 + 54a-2z4 - 25a-2z6 + a-2z8 + a-2z10 + 2a-1z-1 - 5a-1z - 3a-1z3 + 14a-1z5 - 10a-1z7 + 2a-1z9 - z-2 + 6 - 13z2 + 13z4 - 6z6 + z8

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 = 19                       1 j = 17                     2   j = 15                   3 1   j = 13                 4 2     j = 11               6 4       j = 9             5 3         j = 7           4 6           j = 5         5 5             j = 3       4 7               j = 1     1 2                 j = -1   1 4                   j = -3   1                     j = -5 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, 448]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 448]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 4, 15, 3], X[8, 18, 9, 17], X[16, 8, 17, 7], > X[18, 10, 19, 9], X[12, 14, 5, 13], X[22, 20, 13, 19], X[20, 11, 21, 12], > X[10, 21, 11, 22], X[2, 5, 3, 6], X[4, 16, 1, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, -3, 5, -9, 8, -6}, > {6, -2, 11, -4, 3, -5, 7, -8, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-2 2 2 3 4 5 6 7 8 9 5 + q - - - 6 q + 9 q - 9 q + 11 q - 9 q + 7 q - 5 q + 3 q - q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 -4 -2 2 4 6 8 10 12 14 22 3 + q + q + q + 2 q + 4 q + 3 q + 3 q + 5 q + q + 4 q - q + 24 26 > q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 448]][a, z]
Out[8]=	2 2 2 2 7 9 -2 1 2 2 4 z 14 z 15 z 4 4 - -- + -- - -- + z + ----- - ----- + 4 z - ---- + ----- - ----- + z - 6 4 2 4 2 2 2 6 4 2 a a a a z a z a a a 4 4 4 6 6 6 8 4 z 13 z 10 z z 6 z 2 z z > ---- + ----- - ----- - -- + ---- - ---- + -- 6 4 2 6 4 2 4 a a a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 448]][a, z]
Out[9]=	-8 9 13 -2 1 2 2 2 z z 3 z 8 z 6 - a + -- + -- - z - ----- - ----- + ---- + --- + -- + -- - --- - --- - 4 2 4 2 2 2 3 a z 9 7 5 3 a a a z a z a z a a a a 2 2 2 2 3 3 3 3 5 z 2 z 2 z 31 z 41 z z 4 z 3 z 9 z > --- - 13 z - --- + ---- - ----- - ----- + --- - ---- - ---- + ---- + a 10 8 4 2 11 9 7 5 a a a a a a a a 3 3 4 4 4 4 4 5 5 4 z 3 z 4 3 z 7 z 5 z 56 z 54 z 5 z 8 z > ---- - ---- + 13 z + ---- - ---- + ---- + ----- + ----- + ---- - ---- - 3 a 10 8 6 4 2 9 7 a a a a a a a a 5 5 5 6 6 6 6 7 7 3 z 24 z 14 z 6 6 z 14 z 39 z 25 z 6 z 9 z > ---- + ----- + ----- - 6 z + ---- - ----- - ----- - ----- + ---- - ---- - 5 3 a 8 6 4 2 7 5 a a a a a a a a 7 7 8 8 8 9 9 9 10 10 25 z 10 z 8 6 z 6 z z 4 z 6 z 2 z z z > ----- - ----- + z + ---- + ---- + -- + ---- + ---- + ---- + --- + --- 3 a 6 4 2 5 3 a 4 2 a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 5 1 1 1 4 q 2 q 4 q 5 7 7 q + 5 q + ----- + ----- + ---- + ---- + -- + --- + ---- + 5 q t + 4 q t + 5 4 3 3 3 2 2 t t q t q t q t q t t 7 2 9 2 9 3 11 3 11 4 13 4 13 5 > 6 q t + 5 q t + 3 q t + 6 q t + 4 q t + 4 q t + 2 q t + 15 5 15 6 17 6 19 7 > 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a448

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