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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- 4q3/2 + 6q5/2 - 13q7/2 + 11q9/2 - 15q11/2 + 12q13/2 - 10q15/2 + 6q17/2 - 2q19/2 + q21/2

A2 (sl(3)) Invariant:

4q4 + q6 + 8q8 + 12q10 + 12q12 + 20q14 + 15q16 + 16q18 + 9q20 + 2q22 + q24 - 7q26 - 5q28 - 3q30 - 3q32 - q34

HOMFLY-PT Polynomial:

a-11z-3 + a-11z-1 - 5a-9z-3 - 8a-9z-1 - 4a-9z + 9a-7z-3 + 20a-7z-1 + 17a-7z + 6a-7z3 - 7a-5z-3 - 20a-5z-1 - 22a-5z - 12a-5z3 - 3a-5z5 + 2a-3z-3 + 7a-3z-1 + 9a-3z + 4a-3z3

Kauffman Polynomial:

- a-12z-2 + 4a-12 - 6a-12z2 + 4a-12z4 - a-12z6 + a-11z-3 - 2a-11z-1 + 2a-11z - 4a-11z3 + 5a-11z5 - 2a-11z7 - 7a-10z-2 + 24a-10 - 34a-10z2 + 21a-10z4 - 2a-10z6 - 2a-10z8 + 5a-9z-3 - 12a-9z-1 + 16a-9z - 22a-9z3 + 25a-9z5 - 8a-9z7 - a-9z9 - 18a-8z-2 + 58a-8 - 75a-8z2 + 46a-8z4 - 7a-8z8 + 9a-7z-3 - 24a-7z-1 + 37a-7z - 50a-7z3 + 46a-7z5 - 16a-7z7 - a-7z9 - 19a-6z-2 + 60a-6 - 73a-6z2 + 41a-6z4 - 5a-6z6 - 5a-6z8 + 7a-5z-3 - 23a-5z-1 + 39a-5z - 42a-5z3 + 26a-5z5 - 10a-5z7 - 7a-4z-2 + 23a-4 - 26a-4z2 + 12a-4z4 - 6a-4z6 + 2a-3z-3 - 9a-3z-1 + 16a-3z - 10a-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                 1   j = 18               5 1   j = 16             5 1     j = 14           7 5       j = 12         8 5         j = 10       7 11           j = 8     6 4             j = 6     7               j = 4 4 6                 j = 2 4

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, 438]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 438]]
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, 15, 8, 14], X[13, 5, 14, 8], X[19, 13, 20, 22], > X[15, 21, 16, 20], X[21, 17, 22, 16], X[17, 9, 18, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, -3, 11}, > {-7, 6, -9, 10, -11, 3, -8, 9, -10, 8}]
In[6]:=	Jones[L][q]
Out[6]=	3/2 5/2 7/2 9/2 11/2 13/2 15/2 -4 q + 6 q - 13 q + 11 q - 15 q + 12 q - 10 q + 17/2 19/2 21/2 > 6 q - 2 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	4 6 8 10 12 14 16 18 20 22 4 q + q + 8 q + 12 q + 12 q + 20 q + 15 q + 16 q + 9 q + 2 q + 24 26 28 30 32 34 > q - 7 q - 5 q - 3 q - 3 q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 438]][a, z]
Out[8]=	1 5 9 7 2 1 8 20 20 7 ------ - ----- + ----- - ----- + ----- + ----- - ---- + ---- - ---- + ---- - 11 3 9 3 7 3 5 3 3 3 11 9 7 5 3 a z a z a z a z a z a z a z a z a z a z 3 3 3 5 4 z 17 z 22 z 9 z 6 z 12 z 4 z 3 z > --- + ---- - ---- + --- + ---- - ----- + ---- - ---- 9 7 5 3 7 5 3 5 a a a a a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 438]][a, z]
Out[9]=	4 24 58 60 23 1 5 9 7 2 1 --- + --- + -- + -- + -- + ------ + ----- + ----- + ----- + ----- - ------ - 12 10 8 6 4 11 3 9 3 7 3 5 3 3 3 12 2 a a a a a a z a z a z a z a z a z 7 18 19 7 2 12 24 23 9 2 z > ------ - ----- - ----- - ----- - ----- - ---- - ---- - ---- - ---- + --- + 10 2 8 2 6 2 4 2 11 9 7 5 3 11 a z a z a z a z a z a z a z a z a z a 2 2 2 2 2 3 16 z 37 z 39 z 16 z 6 z 34 z 75 z 73 z 26 z 4 z > ---- + ---- + ---- + ---- - ---- - ----- - ----- - ----- - ----- - ---- - 9 7 5 3 12 10 8 6 4 11 a a a a a a a a a a 3 3 3 3 4 4 4 4 4 22 z 50 z 42 z 10 z 4 z 21 z 46 z 41 z 12 z > ----- - ----- - ----- - ----- + ---- + ----- + ----- + ----- + ----- + 9 7 5 3 12 10 8 6 4 a a a a a a a a a 5 5 5 5 6 6 6 6 7 7 5 z 25 z 46 z 26 z z 2 z 5 z 6 z 2 z 8 z > ---- + ----- + ----- + ----- - --- - ---- - ---- - ---- - ---- - ---- - 11 9 7 5 12 10 6 4 11 9 a a a a a a a a a a 7 7 8 8 8 9 9 16 z 10 z 2 z 7 z 5 z z z > ----- - ----- - ---- - ---- - ---- - -- - -- 7 5 10 8 6 9 7 a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 4 4 6 2 8 2 8 3 10 3 10 4 4 q + 4 q + 6 q t + 7 q t + 6 q t + 4 q t + 7 q t + 11 q t + 12 4 12 5 14 5 14 6 16 6 16 7 18 7 > 8 q t + 5 q t + 7 q t + 5 q t + 5 q t + q t + 5 q t + 18 8 20 8 22 9 > q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n438

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