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DrawMorseLink

PD Presentation:

X8192 X14,4,15,3 X10,14,11,13 X22,10,13,9 X20,17,21,18 X18,8,19,7 X12,20,7,19 X16,6,17,5 X2,11,3,12 X4,16,5,15 X6,22,1,21

Gauss Code:

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

Jones Polynomial:

- q-1 + 4 - 7q + 12q2 - 15q3 + 18q4 - 17q5 + 16q6 - 11q7 + 7q8 - 3q9 + q10

A2 (sl(3)) Invariant:

- q-2 + 2 - q2 + 2q4 + 2q6 - q8 + 5q10 - 2q12 + 5q14 + 3q16 + 3q18 + 6q20 + 3q24 + q30

HOMFLY-PT Polynomial:

a-8z-2 + 2a-8 + 3a-8z2 + a-8z4 - 2a-6z-2 - 6a-6 - 10a-6z2 - 8a-6z4 - 2a-6z6 + a-4z-2 + 3a-4 + 8a-4z2 + 9a-4z4 + 5a-4z6 + a-4z8 + a-2 - a-2z2 - 3a-2z4 - a-2z6

Kauffman Polynomial:

- a-12z2 + a-12z4 - 2a-11z3 + 3a-11z5 - a-10 + 5a-10z2 - 7a-10z4 + 6a-10z6 + 5a-9z3 - 10a-9z5 + 8a-9z7 - a-8z-2 + 5a-8 - 12a-8z2 + 17a-8z4 - 17a-8z6 + 9a-8z8 + 2a-7z-1 - 9a-7z + 17a-7z3 - 12a-7z5 - 5a-7z7 + 6a-7z9 - 2a-6z-2 + 11a-6 - 31a-6z2 + 50a-6z4 - 43a-6z6 + 10a-6z8 + 2a-6z10 + 2a-5z-1 - 9a-5z + 7a-5z3 + 16a-5z5 - 30a-5z7 + 11a-5z9 - a-4z-2 + 5a-4 - 17a-4z2 + 41a-4z4 - 35a-4z6 + 5a-4z8 + 2a-4z10 - a-3z3 + 12a-3z5 - 16a-3z7 + 5a-3z9 - a-2 - 4a-2z2 + 16a-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 1 j = 17                   4     j = 15                 7 3     j = 13               9 4       j = 11             8 7         j = 9           10 9           j = 7         7 10             j = 5       5 8               j = 3     3 8                 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, 521]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 521]]
Out[4]=	PD[X[8, 1, 9, 2], X[14, 4, 15, 3], X[10, 14, 11, 13], X[22, 10, 13, 9], > X[20, 17, 21, 18], X[18, 8, 19, 7], X[12, 20, 7, 19], X[16, 6, 17, 5], > X[2, 11, 3, 12], X[4, 16, 5, 15], X[6, 22, 1, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10, 8, -11}, {6, -1, 4, -3, 9, -7}, > {3, -2, 10, -8, 5, -6, 7, -5, 11, -4}]
In[6]:=	Jones[L][q]
Out[6]=	1 2 3 4 5 6 7 8 9 10 4 - - - 7 q + 12 q - 15 q + 18 q - 17 q + 16 q - 11 q + 7 q - 3 q + q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-2 2 4 6 8 10 12 14 16 18 2 - q - q + 2 q + 2 q - q + 5 q - 2 q + 5 q + 3 q + 3 q + 20 24 30 > 6 q + 3 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 521]][a, z]
Out[8]=	2 2 2 2 4 2 6 3 -2 1 2 1 3 z 10 z 8 z z z -- - -- + -- + a + ----- - ----- + ----- + ---- - ----- + ---- - -- + -- - 8 6 4 8 2 6 2 4 2 8 6 4 2 8 a a a a z a z a z a a a a a 4 4 4 6 6 6 8 8 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, 521]][a, z]
Out[9]=	-10 5 11 5 -2 1 2 1 2 2 9 z 9 z -a + -- + -- + -- - a - ----- - ----- - ----- + ---- + ---- - --- - --- - 8 6 4 8 2 6 2 4 2 7 5 7 5 a a a a z a z a z a z a z a a 2 2 2 2 2 2 3 3 3 3 z 5 z 12 z 31 z 17 z 4 z 2 z 5 z 17 z 7 z > --- + ---- - ----- - ----- - ----- - ---- - ---- + ---- + ----- + ---- - 12 10 8 6 4 2 11 9 7 5 a a a a a a a a a a 3 3 4 4 4 4 4 4 5 5 z 2 z z 7 z 17 z 50 z 41 z 16 z 3 z 10 z > -- + ---- + --- - ---- + ----- + ----- + ----- + ----- + ---- - ----- - 3 a 12 10 8 6 4 2 11 9 a a a a a a a a a 5 5 5 5 6 6 6 6 6 12 z 16 z 12 z 3 z 6 z 17 z 43 z 35 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 8 z 5 z 30 z 16 z z 9 z 10 z 5 z 4 z 6 z > ---- - ---- - ----- - ----- + -- + ---- + ----- + ---- + ---- + ---- + 9 7 5 3 a 8 6 4 2 7 a a a a a a a a a 9 9 10 10 11 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 8 q + 5 q + ----- + ---- + -- + --- + ---- + 8 q t + 7 q t + 10 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 15 5 > 10 q t + 9 q t + 8 q t + 7 q t + 9 q t + 4 q t + 7 q t + 15 6 17 6 19 7 19 8 21 8 > 3 q t + 4 q t + 3 q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a521

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