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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- 3q3/2 + 5q5/2 - 11q7/2 + 11q9/2 - 15q11/2 + 12q13/2 - 11q15/2 + 7q17/2 - 4q19/2 + q21/2

A2 (sl(3)) Invariant:

3q4 + 5q8 + 8q10 + 6q12 + 13q14 + 10q16 + 12q18 + 10q20 + 5q22 + 7q24 + q28 + 2q30 - q32

HOMFLY-PT Polynomial:

- a-9z-3 - a-9z-1 + a-9z3 + 3a-7z-3 + 6a-7z-1 + 4a-7z - a-7z5 - 3a-5z-3 - 9a-5z-1 - 10a-5z - 6a-5z3 - 2a-5z5 + a-3z-3 + 4a-3z-1 + 6a-3z + 3a-3z3

Kauffman Polynomial:

- a-12z2 + 2a-12z4 - a-12z6 + 2a-11z - 8a-11z3 + 11a-11z5 - 4a-11z7 - 2a-10z2 - 2a-10z4 + 11a-10z6 - 5a-10z8 + a-9z-3 - 5a-9z-1 + 16a-9z - 32a-9z3 + 33a-9z5 - 7a-9z7 - 2a-9z9 - 3a-8z-2 + 10a-8 - 10a-8z2 - 9a-8z4 + 25a-8z6 - 11a-8z8 + 3a-7z-3 - 12a-7z-1 + 31a-7z - 52a-7z3 + 41a-7z5 - 10a-7z7 - 2a-7z9 - 6a-6z-2 + 19a-6 - 18a-6z2 - 2a-6z4 + 10a-6z6 - 6a-6z8 + 3a-5z-3 - 12a-5z-1 + 27a-5z - 34a-5z3 + 19a-5z5 - 7a-5z7 - 3a-4z-2 + 10a-4 - 9a-4z2 + 3a-4z4 - 3a-4z6 + a-3z-3 - 5a-3z-1 + 10a-3z - 6a-3z3

Khovanov Homology:

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 22                   1 j = 20                 3   j = 18               4 1   j = 16             7 3     j = 14           7 6       j = 12         8 5         j = 10       5 9           j = 8     6 6             j = 6   1 7               j = 4 2 4                 j = 2 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, 458]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 458]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 13, 4, 12], X[7, 21, 8, 20], X[19, 5, 20, 10], > X[13, 19, 14, 22], X[21, 11, 22, 18], X[17, 15, 18, 14], X[9, 17, 10, 16], > X[15, 9, 16, 8], X[2, 5, 3, 6], X[11, 1, 12, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -2, 11}, {-4, 3, -6, 5}, {10, -1, -3, 9, -8, 4}, > {-11, 2, -5, 7, -9, 8, -7, 6}]
In[6]:=	Jones[L][q]
Out[6]=	3/2 5/2 7/2 9/2 11/2 13/2 15/2 -3 q + 5 q - 11 q + 11 q - 15 q + 12 q - 11 q + 17/2 19/2 21/2 > 7 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	4 8 10 12 14 16 18 20 22 3 q + 5 q + 8 q + 6 q + 13 q + 10 q + 12 q + 10 q + 5 q + 24 28 30 32 > 7 q + q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 458]][a, z]
Out[8]=	1 3 3 1 1 6 9 4 4 z 10 z -(-----) + ----- - ----- + ----- - ---- + ---- - ---- + ---- + --- - ---- + 9 3 7 3 5 3 3 3 9 7 5 3 7 5 a z a z a z a z a z a z a z a z a a 3 3 3 5 5 6 z z 6 z 3 z z 2 z > --- + -- - ---- + ---- - -- - ---- 3 9 5 3 7 5 a a a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 458]][a, z]
Out[9]=	10 19 10 1 3 3 1 3 6 3 5 -- + -- + -- + ----- + ----- + ----- + ----- - ----- - ----- - ----- - ---- - 8 6 4 9 3 7 3 5 3 3 3 8 2 6 2 4 2 9 a a a a z a z a z a z a z a z a z a z 2 2 2 12 12 5 2 z 16 z 31 z 27 z 10 z z 2 z 10 z > ---- - ---- - ---- + --- + ---- + ---- + ---- + ---- - --- - ---- - ----- - 7 5 3 11 9 7 5 3 12 10 8 a z a z a z a a a a a a a a 2 2 3 3 3 3 3 4 4 4 18 z 9 z 8 z 32 z 52 z 34 z 6 z 2 z 2 z 9 z > ----- - ---- - ---- - ----- - ----- - ----- - ---- + ---- - ---- - ---- - 6 4 11 9 7 5 3 12 10 8 a a a a a a a a a a 4 4 5 5 5 5 6 6 6 6 2 z 3 z 11 z 33 z 41 z 19 z z 11 z 25 z 10 z > ---- + ---- + ----- + ----- + ----- + ----- - --- + ----- + ----- + ----- - 6 4 11 9 7 5 12 10 8 6 a a a a a a a a a a 6 7 7 7 7 8 8 8 9 9 3 z 4 z 7 z 10 z 7 z 5 z 11 z 6 z 2 z 2 z > ---- - ---- - ---- - ----- - ---- - ---- - ----- - ---- - ---- - ---- 4 11 9 7 5 10 8 6 9 7 a a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 4 4 6 6 2 8 2 8 3 10 3 3 q + 2 q + 4 q t + q t + 7 q t + 6 q t + 6 q t + 5 q t + 10 4 12 4 12 5 14 5 14 6 16 6 > 9 q t + 8 q t + 5 q t + 7 q t + 6 q t + 7 q t + 16 7 18 7 18 8 20 8 22 9 > 3 q t + 4 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n458

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