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DrawMorseLink

PD Presentation:

X6172 X2,16,3,15 X10,4,11,3 X14,6,15,5 X22,12,13,11 X12,14,5,13 X4,21,1,22 X20,17,21,18 X16,7,17,8 X8,20,9,19 X18,10,19,9

Gauss Code:

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

Jones Polynomial:

q-3 - 4q-2 + 8q-1 - 12 + 18q - 20q2 + 21q3 - 17q4 + 14q5 - 8q6 + 4q7 - q8

A2 (sl(3)) Invariant:

q-8 - 2q-6 + 2q-4 - q-2 + 4q2 - 3q4 + 6q6 + q8 + 5q10 + 6q12 + q14 + 6q16 + q20 + q22 - q24

HOMFLY-PT Polynomial:

a-6z-2 - 2a-6z2 - a-6z4 - 2a-4z-2 - a-4 + 6a-4z2 + 7a-4z4 + 2a-4z6 + a-2z-2 - 6a-2z2 - 9a-2z4 - 5a-2z6 - a-2z8 + 1 + 2z2 + 3z4 + z6

Kauffman Polynomial:

- a-9z3 + a-9z5 + 2a-8z2 - 6a-8z4 + 4a-8z6 + 2a-7z3 - 10a-7z5 + 7a-7z7 + a-6z-2 - 3a-6 + a-6z2 + 3a-6z4 - 11a-6z6 + 8a-6z8 - 2a-5z-1 + 4a-5z - a-5z3 - a-5z5 - 5a-5z7 + 6a-5z9 + 2a-4z-2 - 4a-4 - 11a-4z2 + 37a-4z4 - 35a-4z6 + 11a-4z8 + 2a-4z10 - 2a-3z-1 + 4a-3z - 7a-3z3 + 24a-3z5 - 28a-3z7 + 12a-3z9 + a-2z-2 - a-2 - 16a-2z2 + 47a-2z4 - 41a-2z6 + 10a-2z8 + 2a-2z10 + a-1z3 + 4a-1z5 - 12a-1z7 + 6a-1z9 + 1 - 6z2 + 17z4 - 20z6 + 7z8 + 4az3 - 10az5 + 4az7 - 2a2z4 + a2z6

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 = 17                       1 j = 15                     3   j = 13                   5 1   j = 11                 9 3     j = 9               10 7       j = 7             11 7         j = 5           9 10           j = 3         9 11             j = 1       5 11               j = -1     3 7                 j = -3   1 5                   j = -5   3                     j = -7 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, Alternating, 438]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 438]]
Out[4]=	PD[X[6, 1, 7, 2], X[2, 16, 3, 15], X[10, 4, 11, 3], X[14, 6, 15, 5], > X[22, 12, 13, 11], X[12, 14, 5, 13], X[4, 21, 1, 22], X[20, 17, 21, 18], > X[16, 7, 17, 8], X[8, 20, 9, 19], X[18, 10, 19, 9]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -7}, {4, -1, 9, -10, 11, -3, 5, -6}, > {6, -4, 2, -9, 8, -11, 10, -8, 7, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-3 4 8 2 3 4 5 6 7 8 -12 + q - -- + - + 18 q - 20 q + 21 q - 17 q + 14 q - 8 q + 4 q - q 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 2 2 -2 2 4 6 8 10 12 14 16 q - -- + -- - q + 4 q - 3 q + 6 q + q + 5 q + 6 q + q + 6 q + 6 4 q q 20 22 24 > q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 438]][a, z]
Out[8]=	2 2 2 4 -4 1 2 1 2 2 z 6 z 6 z 4 z 1 - a + ----- - ----- + ----- + 2 z - ---- + ---- - ---- + 3 z - -- + 6 2 4 2 2 2 6 4 2 6 a z a z a z a a a a 4 4 6 6 8 7 z 9 z 6 2 z 5 z z > ---- - ---- + z + ---- - ---- - -- 4 2 4 2 2 a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 438]][a, z]
Out[9]=	3 4 -2 1 2 1 2 2 4 z 4 z 2 1 - -- - -- - a + ----- + ----- + ----- - ---- - ---- + --- + --- - 6 z + 6 4 6 2 4 2 2 2 5 3 5 3 a a a z a z a z a z a z a a 2 2 2 2 3 3 3 3 3 2 z z 11 z 16 z z 2 z z 7 z z 3 4 > ---- + -- - ----- - ----- - -- + ---- - -- - ---- + -- + 4 a z + 17 z - 8 6 4 2 9 7 5 3 a a a a a a a a a 4 4 4 4 5 5 5 5 5 6 z 3 z 37 z 47 z 2 4 z 10 z z 24 z 4 z > ---- + ---- + ----- + ----- - 2 a z + -- - ----- - -- + ----- + ---- - 8 6 4 2 9 7 5 3 a a a a a a a a a 6 6 6 6 7 7 5 6 4 z 11 z 35 z 41 z 2 6 7 z 5 z > 10 a z - 20 z + ---- - ----- - ----- - ----- + a z + ---- - ---- - 8 6 4 2 7 5 a a a a a a 7 7 8 8 8 9 9 28 z 12 z 7 8 8 z 11 z 10 z 6 z 12 z > ----- - ----- + 4 a z + 7 z + ---- + ----- + ----- + ---- + ----- + 3 a 6 4 2 5 3 a a a a a a 9 10 10 6 z 2 z 2 z > ---- + ----- + ----- a 4 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 3 1 5 3 7 5 q 3 11 q + 9 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 11 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 9 q t + 10 q t + 11 q t + 7 q t + 10 q t + 7 q t + 9 q t + 11 5 13 5 13 6 15 6 17 7 > 3 q t + 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a438

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