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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

3q - 5q2 + 9q3 - 8q4 + 11q5 - 9q6 + 7q7 - 5q8 + 2q9 - q10

A2 (sl(3)) Invariant:

3q2 + 2q6 + 6q8 + 4q10 + 8q12 + 6q14 + 5q16 + 4q18 - q20 + q22 - 3q24 - 4q26 - q28 - 2q30 - q32

HOMFLY-PT Polynomial:

- a-10z-2 - a-10 + 3a-8z-2 + 5a-8 + 3a-8z2 - 2a-6z-2 - 5a-6 - 4a-6z2 - 2a-6z4 - a-4z-2 - 2a-4 - 3a-4z2 - 2a-4z4 + a-2z-2 + 3a-2 + 3a-2z2

Kauffman Polynomial:

2a-11z-1 - 7a-11z + 9a-11z3 - 5a-11z5 + a-11z7 - a-10z-2 + a-10 - 2a-10z2 + 9a-10z4 - 8a-10z6 + 2a-10z8 + 8a-9z-1 - 27a-9z + 43a-9z3 - 26a-9z5 + 3a-9z7 + a-9z9 - 3a-8z-2 + 5a-8 - 8a-8z2 + 20a-8z4 - 23a-8z6 + 7a-8z8 + 10a-7z-1 - 34a-7z + 52a-7z3 - 43a-7z5 + 10a-7z7 + a-7z9 - 2a-6z-2 + 4a-6 - 3a-6z2 - a-6z4 - 8a-6z6 + 5a-6z8 + 2a-5z-1 - 10a-5z + 18a-5z3 - 19a-5z5 + 8a-5z7 + a-4z-2 - 3a-4 + 9a-4z2 - 12a-4z4 + 7a-4z6 - 2a-3z-1 + 4a-3z + 3a-3z5 + a-2z-2 - 4a-2 + 6a-2z2

Khovanov Homology:

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 21                   1 j = 19                 1   j = 17               4 1   j = 15             3 1     j = 13           6 4       j = 11         5 3         j = 9       4 7           j = 7     5 4             j = 5     4               j = 3 3 5                 j = 1 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 271]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 271]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 11, 4, 10], X[7, 17, 8, 16], X[15, 5, 16, 8], > X[11, 19, 12, 18], X[17, 9, 18, 22], X[21, 13, 22, 12], X[13, 21, 14, 20], > X[19, 15, 20, 14], X[2, 5, 3, 6], X[9, 1, 10, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -2, 11}, {10, -1, -3, 4}, > {-11, 2, -5, 7, -8, 9, -4, 3, -6, 5, -9, 8, -7, 6}]
In[6]:=	Jones[L][q]
Out[6]=	2 3 4 5 6 7 8 9 10 3 q - 5 q + 9 q - 8 q + 11 q - 9 q + 7 q - 5 q + 2 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 6 8 10 12 14 16 18 20 22 3 q + 2 q + 6 q + 4 q + 8 q + 6 q + 5 q + 4 q - q + q - 24 26 28 30 32 > 3 q - 4 q - q - 2 q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 271]][a, z]
Out[8]=	2 -10 5 5 2 3 1 3 2 1 1 3 z -a + -- - -- - -- + -- - ------ + ----- - ----- - ----- + ----- + ---- - 8 6 4 2 10 2 8 2 6 2 4 2 2 2 8 a a a a a z a z a z a z a z a 2 2 2 4 4 4 z 3 z 3 z 2 z 2 z > ---- - ---- + ---- - ---- - ---- 6 4 2 6 4 a a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 271]][a, z]
Out[9]=	-10 5 4 3 4 1 3 2 1 1 2 a + -- + -- - -- - -- - ------ - ----- - ----- + ----- + ----- + ----- + 8 6 4 2 10 2 8 2 6 2 4 2 2 2 11 a a a a a z a z a z a z a z a z 2 2 8 10 2 2 7 z 27 z 34 z 10 z 4 z 2 z 8 z > ---- + ---- + ---- - ---- - --- - ---- - ---- - ---- + --- - ---- - ---- - 9 7 5 3 11 9 7 5 3 10 8 a z a z a z a z a a a a a a a 2 2 2 3 3 3 3 4 4 4 3 z 9 z 6 z 9 z 43 z 52 z 18 z 9 z 20 z z > ---- + ---- + ---- + ---- + ----- + ----- + ----- + ---- + ----- - -- - 6 4 2 11 9 7 5 10 8 6 a a a a a a a a a a 4 5 5 5 5 5 6 6 6 6 12 z 5 z 26 z 43 z 19 z 3 z 8 z 23 z 8 z 7 z > ----- - ---- - ----- - ----- - ----- + ---- - ---- - ----- - ---- + ---- + 4 11 9 7 5 3 10 8 6 4 a a a a a a a a a a 7 7 7 7 8 8 8 9 9 z 3 z 10 z 8 z 2 z 7 z 5 z z z > --- + ---- + ----- + ---- + ---- + ---- + ---- + -- + -- 11 9 7 5 10 8 6 9 7 a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 5 2 7 2 7 3 9 3 9 4 3 q + 3 q + 5 q t + 4 q t + 5 q t + 4 q t + 4 q t + 7 q t + 11 4 11 5 13 5 13 6 15 6 15 7 17 7 > 5 q t + 3 q t + 6 q t + 4 q t + 3 q t + q t + 4 q t + 17 8 19 8 21 9 > q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n271

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