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DrawMorseLink

PD Presentation:

X6172 X2536 X18,11,19,12 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 X12,17,9,18

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:

- q-5/2 - q-1/2 - 5q1/2 + 3q3/2 - 7q5/2 + 5q7/2 - 6q9/2 + 5q11/2 - 2q13/2 + q15/2

A2 (sl(3)) Invariant:

q-8 + 2q-6 + 5q-4 + 7q-2 + 14 + 16q2 + 17q4 + 18q6 + 11q8 + 8q10 - 3q14 - 4q16 - 6q18 - 2q20 - 2q22 - q24

HOMFLY-PT Polynomial:

a-7z-3 + 2a-7z-1 + a-7z - 5a-5z-3 - 11a-5z-1 - 9a-5z - 3a-5z3 + 9a-3z-3 + 21a-3z-1 + 21a-3z + 10a-3z3 + 2a-3z5 - 7a-1z-3 - 17a-1z-1 - 17a-1z - 8a-1z3 - a-1z5 + 2az-3 + 5az-1 + 4az + az3

Kauffman Polynomial:

- a-8z-2 + 4a-8 - 6a-8z2 + 4a-8z4 - a-8z6 + a-7z-3 - 2a-7z-1 + 2a-7z - 4a-7z3 + 6a-7z5 - 2a-7z7 - 7a-6z-2 + 24a-6 - 34a-6z2 + 26a-6z4 - 4a-6z6 - a-6z8 + 5a-5z-3 - 12a-5z-1 + 16a-5z - 23a-5z3 + 24a-5z5 - 7a-5z7 - 18a-4z-2 + 58a-4 - 75a-4z2 + 43a-4z4 - 7a-4z6 - a-4z8 + 9a-3z-3 - 24a-3z-1 + 37a-3z - 41a-3z3 + 23a-3z5 - 5a-3z7 - 19a-2z-2 + 60a-2 - 73a-2z2 + 32a-2z4 - 5a-2z6 + 7a-1z-3 - 23a-1z-1 + 39a-1z - 37a-1z3 + 12a-1z5 - a-1z7 - 7z-2 + 23 - 26z2 + 11z4 - z6 + 2az-3 - 9az-1 + 16az - 15az3 + 7az5 - az7

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 = 16                       1 j = 14                     1   j = 12                   4 1   j = 10                 2 1     j = 8               4 5       j = 6           1 4 1         j = 4             4           j = 2       1 7 4             j = 0         6               j = -2     1                   j = -4 1                       j = -6 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]=	4
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 440]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 440]]
Out[4]=	PD[X[6, 1, 7, 2], X[2, 5, 3, 6], X[18, 11, 19, 12], 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[12, 17, 9, 18]]
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]=	-(5/2) 1 3/2 5/2 7/2 9/2 11/2 -q - ------- - 5 Sqrt[q] + 3 q - 7 q + 5 q - 6 q + 5 q - Sqrt[q] 13/2 15/2 > 2 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 2 5 7 2 4 6 8 10 14 14 + q + -- + -- + -- + 16 q + 17 q + 18 q + 11 q + 8 q - 3 q - 6 4 2 q q q 16 18 20 22 24 > 4 q - 6 q - 2 q - 2 q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 440]][a, z]
Out[8]=	1 5 9 7 2 a 2 11 21 17 5 a z ----- - ----- + ----- - ---- + --- + ---- - ---- + ---- - --- + --- + -- - 7 3 5 3 3 3 3 3 7 5 3 a z z 7 a z a z a z a z z a z a z a z a 3 3 3 5 5 9 z 21 z 17 z 3 z 10 z 8 z 3 2 z z > --- + ---- - ---- + 4 a z - ---- + ----- - ---- + a z + ---- - -- 5 3 a 5 3 a 3 a a a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 440]][a, z]
Out[9]=	4 24 58 60 1 5 9 7 2 a 7 1 23 + -- + -- + -- + -- + ----- + ----- + ----- + ---- + --- - -- - ----- - 8 6 4 2 7 3 5 3 3 3 3 3 2 8 2 a a a a a z a z a z a z z z a z 7 18 19 2 12 24 23 9 a 2 z 16 z > ----- - ----- - ----- - ---- - ---- - ---- - --- - --- + --- + ---- + 6 2 4 2 2 2 7 5 3 a z z 7 5 a z a z a z a z a z a z a a 2 2 2 2 3 37 z 39 z 2 6 z 34 z 75 z 73 z 4 z > ---- + ---- + 16 a z - 26 z - ---- - ----- - ----- - ----- - ---- - 3 a 8 6 4 2 7 a a a a a a 3 3 3 4 4 4 4 23 z 41 z 37 z 3 4 4 z 26 z 43 z 32 z > ----- - ----- - ----- - 15 a z + 11 z + ---- + ----- + ----- + ----- + 5 3 a 8 6 4 2 a a a a a a 5 5 5 5 6 6 6 6 6 z 24 z 23 z 12 z 5 6 z 4 z 7 z 5 z > ---- + ----- + ----- + ----- + 7 a z - z - -- - ---- - ---- - ---- - 7 5 3 a 8 6 4 2 a a a a a a a 7 7 7 7 8 8 2 z 7 z 5 z z 7 z z > ---- - ---- - ---- - -- - a z - -- - -- 7 5 3 a 6 4 a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 1 1 q 2 6 4 2 6 2 6 + 7 q + ----- + ----- + ----- + -- + 4 q t + q t + 4 q t + 4 q t + 6 4 4 4 2 2 t q t q t q t 6 3 8 3 8 4 10 4 10 5 12 5 12 6 > q t + 4 q t + 5 q t + 2 q t + q t + 4 q t + q t + 14 6 16 7 > q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n440

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