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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-1 + 4 - 7q + 13q2 - 16q3 + 21q4 - 20q5 + 18q6 - 14q7 + 9q8 - 4q9 + q10

A2 (sl(3)) Invariant:

- q-2 + 2 - q2 + 3q4 + 4q6 + q8 + 9q10 + q12 + 6q14 + 2q16 - q18 + 3q20 - 3q22 + 2q24 - q28 + q30

HOMFLY-PT Polynomial:

a-8 + 2a-8z2 + a-8z4 + a-6z-2 - 2a-6 - 7a-6z2 - 7a-6z4 - 2a-6z6 - 2a-4z-2 - a-4 + 6a-4z2 + 9a-4z4 + 5a-4z6 + a-4z8 + a-2z-2 + 2a-2 - a-2z2 - 3a-2z4 - a-2z6

Kauffman Polynomial:

a-12z4 - a-11z3 + 4a-11z5 - a-10 + 3a-10z2 - 8a-10z4 + 9a-10z6 - 2a-9z + 9a-9z3 - 18a-9z5 + 13a-9z7 - a-8 + 4a-8z2 - 15a-8z6 + 12a-8z8 - 6a-7z + 21a-7z3 - 27a-7z5 + a-7z7 + 7a-7z9 + a-6z-2 - a-6 - 9a-6z2 + 38a-6z4 - 49a-6z6 + 15a-6z8 + 2a-6z10 - 2a-5z-1 - 2a-5z + 10a-5z3 + 8a-5z5 - 28a-5z7 + 12a-5z9 + 2a-4z-2 - 2a-4 - 15a-4z2 + 46a-4z4 - 40a-4z6 + 7a-4z8 + 2a-4z10 - 2a-3z-1 + 2a-3z + a-3z3 + 10a-3z5 - 15a-3z7 + 5a-3z9 + a-2z-2 - 2a-2 - 5a-2z2 + 17a-2z4 - 15a-2z6 + 4a-2z8 + 2a-1z3 - 3a-1z5 + a-1z7

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 21                       1 j = 19                     3   j = 17                   6 1   j = 15                 8 3     j = 13               10 6       j = 11             12 10         j = 9           9 8           j = 7         7 12             j = 5       6 9               j = 3     3 9                 j = 1   1 4                   j = -1   3                     j = -3 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, 439]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 439]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 6, 15, 5], X[10, 4, 11, 3], X[2, 16, 3, 15], > X[20, 18, 21, 17], X[16, 9, 17, 10], X[8, 20, 9, 19], X[18, 8, 19, 7], > X[22, 12, 13, 11], X[12, 14, 5, 13], X[4, 21, 1, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -11}, {2, -1, 8, -7, 6, -3, 9, -10}, > {10, -2, 4, -6, 5, -8, 7, -5, 11, -9}]
In[6]:=	Jones[L][q]
Out[6]=	1 2 3 4 5 6 7 8 9 10 4 - - - 7 q + 13 q - 16 q + 21 q - 20 q + 18 q - 14 q + 9 q - 4 q + q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-2 2 4 6 8 10 12 14 16 18 20 2 - q - q + 3 q + 4 q + q + 9 q + q + 6 q + 2 q - q + 3 q - 22 24 28 30 > 3 q + 2 q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 439]][a, z]
Out[8]=	2 2 2 2 4 -8 2 -4 2 1 2 1 2 z 7 z 6 z z z a - -- - a + -- + ----- - ----- + ----- + ---- - ---- + ---- - -- + -- - 6 2 6 2 4 2 2 2 8 6 4 2 8 a a a z a z a z a a a a a 4 4 4 6 6 6 8 7 z 9 z 3 z 2 z 5 z z z > ---- + ---- - ---- - ---- + ---- - -- + -- 6 4 2 6 4 2 4 a a a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 439]][a, z]
Out[9]=	-10 -8 -6 2 2 1 2 1 2 2 2 z 6 z -a - a - a - -- - -- + ----- + ----- + ----- - ---- - ---- - --- - --- - 4 2 6 2 4 2 2 2 5 3 9 7 a a a z a z a z a z a z a a 2 2 2 2 2 3 3 3 2 z 2 z 3 z 4 z 9 z 15 z 5 z z 9 z 21 z > --- + --- + ---- + ---- - ---- - ----- - ---- - --- + ---- + ----- + 5 3 10 8 6 4 2 11 9 7 a a a a a a a a a a 3 3 3 4 4 4 4 4 5 5 10 z z 2 z z 8 z 38 z 46 z 17 z 4 z 18 z > ----- + -- + ---- + --- - ---- + ----- + ----- + ----- + ---- - ----- - 5 3 a 12 10 6 4 2 11 9 a a a a a a a a a 5 5 5 5 6 6 6 6 6 27 z 8 z 10 z 3 z 9 z 15 z 49 z 40 z 15 z > ----- + ---- + ----- - ---- + ---- - ----- - ----- - ----- - ----- + 7 5 3 a 10 8 6 4 2 a a a a a a a a 7 7 7 7 7 8 8 8 8 9 13 z z 28 z 15 z z 12 z 15 z 7 z 4 z 7 z > ----- + -- - ----- - ----- + -- + ----- + ----- + ---- + ---- + ---- + 9 7 5 3 a 8 6 4 2 7 a a a a a a a a a 9 9 10 10 12 z 5 z 2 z 2 z > ----- + ---- + ----- + ----- 5 3 6 4 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 5 1 3 q 4 q 3 q 5 7 7 2 9 q + 6 q + ----- + ---- + -- + --- + ---- + 9 q t + 7 q t + 12 q t + 3 3 2 2 t t q t q t t 9 2 9 3 11 3 11 4 13 4 13 5 > 9 q t + 8 q t + 12 q t + 10 q t + 10 q t + 6 q t + 15 5 15 6 17 6 17 7 19 7 21 8 > 8 q t + 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a439

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