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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-2 - 2q-1 + 6 - 7q + 12q2 - 12q3 + 13q4 - 12q5 + 9q6 - 6q7 + 3q8 - q9

A2 (sl(3)) Invariant:

q-6 + q-4 + 2q-2 + 5 + 4q2 + 8q4 + 6q6 + 4q8 + 4q10 - 3q12 + q14 - 3q16 - q18 - 2q22 + q24 - q26

HOMFLY-PT Polynomial:

- a-6z-2 - 3a-6 - 5a-6z2 - 4a-6z4 - a-6z6 + 4a-4z-2 + 11a-4 + 17a-4z2 + 14a-4z4 + 6a-4z6 + a-4z8 - 5a-2z-2 - 13a-2 - 17a-2z2 - 10a-2z4 - 2a-2z6 + 2z-2 + 5 + 4z2 + z4

Kauffman Polynomial:

a-11z3 + 3a-10z4 + a-9z - 4a-9z3 + 6a-9z5 + 3a-8z2 - 12a-8z4 + 9a-8z6 + a-7z-1 - 4a-7z + 12a-7z3 - 23a-7z5 + 11a-7z7 - a-6z-2 + 5a-6 - 12a-6z2 + 19a-6z4 - 26a-6z6 + 10a-6z8 + 5a-5z-1 - 21a-5z + 43a-5z3 - 30a-5z5 - 4a-5z7 + 5a-5z9 - 4a-4z-2 + 18a-4 - 52a-4z2 + 87a-4z4 - 62a-4z6 + 12a-4z8 + a-4z10 + 9a-3z-1 - 29a-3z + 31a-3z3 + 9a-3z5 - 24a-3z7 + 7a-3z9 - 5a-2z-2 + 21a-2 - 53a-2z2 + 67a-2z4 - 33a-2z6 + 3a-2z8 + a-2z10 + 5a-1z-1 - 13a-1z + 5a-1z3 + 10a-1z5 - 9a-1z7 + 2a-1z9 - 2z-2 + 9 - 16z2 + 14z4 - 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                   4 1   j = 13                 5 2     j = 11               7 4       j = 9             6 5         j = 7           7 8           j = 5         5 5             j = 3       4 9               j = 1     2 3                 j = -1   1 5                   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, 410]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 410]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[20, 16, 21, 15], X[14, 8, 15, 7], > X[10, 12, 5, 11], X[22, 18, 11, 17], X[16, 22, 17, 21], X[8, 19, 9, 20], > X[18, 9, 19, 10], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, -8, 9, -5}, > {5, -2, 11, -4, 3, -7, 6, -9, 8, -3, 7, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-2 2 2 3 4 5 6 7 8 9 6 + q - - - 7 q + 12 q - 12 q + 13 q - 12 q + 9 q - 6 q + 3 q - q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 -4 2 2 4 6 8 10 12 14 16 5 + q + q + -- + 4 q + 8 q + 6 q + 4 q + 4 q - 3 q + q - 3 q - 2 q 18 22 24 26 > q - 2 q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 410]][a, z]
Out[8]=	2 2 2 3 11 13 2 1 4 5 2 5 z 17 z 17 z 5 - -- + -- - -- + -- - ----- + ----- - ----- + 4 z - ---- + ----- - ----- + 6 4 2 2 6 2 4 2 2 2 6 4 2 a a a z a z a z a z a a a 4 4 4 6 6 6 8 4 4 z 14 z 10 z z 6 z 2 z z > z - ---- + ----- - ----- - -- + ---- - ---- + -- 6 4 2 6 4 2 4 a a a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 410]][a, z]
Out[9]=	5 18 21 2 1 4 5 1 5 9 5 z 9 + -- + -- + -- - -- - ----- - ----- - ----- + ---- + ---- + ---- + --- + -- - 6 4 2 2 6 2 4 2 2 2 7 5 3 a z 9 a a a z a z a z a z a z a z a z a 2 2 2 2 3 4 z 21 z 29 z 13 z 2 3 z 12 z 52 z 53 z z > --- - ---- - ---- - ---- - 16 z + ---- - ----- - ----- - ----- + --- - 7 5 3 a 8 6 4 2 11 a a a a a a a a 3 3 3 3 3 4 4 4 4 z 12 z 43 z 31 z 5 z 4 3 z 12 z 19 z > ---- + ----- + ----- + ----- + ---- + 14 z + ---- - ----- + ----- + 9 7 5 3 a 10 8 6 a a a a a a a 4 4 5 5 5 5 5 6 6 87 z 67 z 6 z 23 z 30 z 9 z 10 z 6 9 z 26 z > ----- + ----- + ---- - ----- - ----- + ---- + ----- - 6 z + ---- - ----- - 4 2 9 7 5 3 a 8 6 a a a a a a a a 6 6 7 7 7 7 8 8 8 62 z 33 z 11 z 4 z 24 z 9 z 8 10 z 12 z 3 z > ----- - ----- + ----- - ---- - ----- - ---- + z + ----- + ----- + ---- + 4 2 7 5 3 a 6 4 2 a a a a a a a a 9 9 9 10 10 5 z 7 z 2 z z z > ---- + ---- + ---- + --- + --- 5 3 a 4 2 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 5 1 1 1 5 2 q 3 q 4 q 5 9 q + 5 q + ----- + ----- + ---- + ---- + --- + --- + ---- + 5 q t + 5 4 3 3 3 2 2 t t q t q t q t q t t 7 7 2 9 2 9 3 11 3 11 4 13 4 > 7 q t + 8 q t + 6 q t + 5 q t + 7 q t + 4 q t + 5 q t + 13 5 15 5 15 6 17 6 19 7 > 2 q t + 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a410

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