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DrawMorseLink

PD Presentation:

X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X18,12,19,11 X22,20,9,19 X20,16,21,15 X16,22,17,21 X12,18,13,17 X2536 X9,1,10,4

Gauss Code:

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

Jones Polynomial:

1 - 2q + 5q2 - 3q3 + 5q4 - 3q5 + 3q6 - q7 - q8 + q9 - q10

A2 (sl(3)) Invariant:

1 + 2q4 + 3q6 + 4q8 + 7q10 + 5q12 + 8q14 + 4q16 + 3q18 + q20 - 2q22 - q24 - 3q26 - 2q28 - 2q30 - q32

HOMFLY-PT Polynomial:

- a-10z-2 - a-10 + 3a-8z-2 + 5a-8 + 2a-8z2 - 2a-6z-2 - 5a-6 - 2a-6z2 - a-4z-2 - 2a-4 - 4a-4z2 - 4a-4z4 - a-4z6 + a-2z-2 + 3a-2 + 3a-2z2 + a-2z4

Kauffman Polynomial:

2a-11z-1 - 7a-11z + 10a-11z3 - 6a-11z5 + a-11z7 - a-10z-2 + a-10 - 3a-10z2 + 8a-10z4 - 6a-10z6 + a-10z8 + 8a-9z-1 - 27a-9z + 37a-9z3 - 17a-9z5 + 2a-9z7 - 3a-8z-2 + 5a-8 - 7a-8z2 + 15a-8z4 - 8a-8z6 + a-8z8 + 10a-7z-1 - 34a-7z + 42a-7z3 - 20a-7z5 + 3a-7z7 - 2a-6z-2 + 4a-6 + 2a-6z2 - 3a-6z4 - 2a-6z6 + a-6z8 + 2a-5z-1 - 10a-5z + 16a-5z3 - 15a-5z5 + 4a-5z7 + a-4z-2 - 3a-4 + 12a-4z2 - 14a-4z4 + a-4z6 + a-4z8 - 2a-3z-1 + 4a-3z + a-3z3 - 6a-3z5 + 2a-3z7 + a-2z-2 - 4a-2 + 6a-2z2 - 4a-2z4 + a-2z6

Khovanov Homology:

trqj r = -2 r = -1 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                         j = 17                   1 1   j = 15               3 1       j = 13             1   1       j = 11           4 5 1         j = 9         2 1 1           j = 7       1 4 1             j = 5     4 2                 j = 3   1 4                   j = 1   1                     j = -1 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, NonAlternating, 261]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 261]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 11, 4, 10], X[7, 15, 8, 14], X[13, 5, 14, 8], > X[18, 12, 19, 11], X[22, 20, 9, 19], X[20, 16, 21, 15], X[16, 22, 17, 21], > X[12, 18, 13, 17], 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, -9, -4, 3, 7, -8, 9, -5, 6, -7, 8, -6}]
In[6]:=	Jones[L][q]
Out[6]=	2 3 4 5 6 7 8 9 10 1 - 2 q + 5 q - 3 q + 5 q - 3 q + 3 q - q - q + q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	4 6 8 10 12 14 16 18 20 22 1 + 2 q + 3 q + 4 q + 7 q + 5 q + 8 q + 4 q + 3 q + q - 2 q - 24 26 28 30 32 > q - 3 q - 2 q - 2 q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 261]][a, z]
Out[8]=	2 -10 5 5 2 3 1 3 2 1 1 2 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 6 2 z 4 z 3 z 4 z z z > ---- - ---- + ---- - ---- + -- - -- 6 4 2 4 2 4 a a a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 261]][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 3 z 7 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 3 4 4 2 z 12 z 6 z 10 z 37 z 42 z 16 z z 8 z 15 z > ---- + ----- + ---- + ----- + ----- + ----- + ----- + -- + ---- + ----- - 6 4 2 11 9 7 5 3 10 8 a a a a a a a a a a 4 4 4 5 5 5 5 5 6 6 3 z 14 z 4 z 6 z 17 z 20 z 15 z 6 z 6 z 8 z > ---- - ----- - ---- - ---- - ----- - ----- - ----- - ---- - ---- - ---- - 6 4 2 11 9 7 5 3 10 8 a a a a a a a a a a 6 6 6 7 7 7 7 7 8 8 8 8 2 z z z z 2 z 3 z 4 z 2 z z z z z > ---- + -- + -- + --- + ---- + ---- + ---- + ---- + --- + -- + -- + -- 6 4 2 11 9 7 5 3 10 8 6 4 a a a a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 5 1 q q 5 7 7 2 9 2 7 3 4 q + 4 q + ---- + - + -- + 2 q t + q t + 4 q t + 2 q t + q t + 2 t t q t 9 3 11 3 9 4 11 4 13 4 11 5 15 5 13 6 > q t + 4 q t + q t + 5 q t + q t + q t + 3 q t + q t + 15 6 17 7 17 8 21 9 > q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n261

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