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Knotscape

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DrawMorseLink

PD Presentation:

X12,1,13,2 X16,8,17,7 X5,1,6,10 X3746 X9,5,10,4 X18,14,19,13 X20,16,11,15 X14,20,15,19 X2,11,3,12 X8,18,9,17

Gauss Code:

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

Jones Polynomial:

- q1/2 + q3/2 - 4q5/2 + 3q7/2 - 4q9/2 + 4q11/2 - 3q13/2 + 3q15/2 - q17/2

A2 (sl(3)) Invariant:

q2 + q4 + 3q6 + 4q8 + 4q10 + 4q12 - 4q18 - 3q20 - 2q22 - q24 + q26 + q30

HOMFLY-PT Polynomial:

- a-9z-1 - a-9z + 5a-7z-1 + 8a-7z + 5a-7z3 + a-7z5 - 8a-5z-1 - 15a-5z - 13a-5z3 - 6a-5z5 - a-5z7 + 4a-3z-1 + 8a-3z + 5a-3z3 + a-3z5

Kauffman Polynomial:

- a-11z + 2a-10 - 3a-10z2 - a-9z-1 - a-9z + 4a-9z3 - 2a-9z5 + 9a-8 - 23a-8z2 + 20a-8z4 - 5a-8z6 - 5a-7z-1 + 11a-7z - 18a-7z3 + 16a-7z5 - 4a-7z7 + 14a-6 - 33a-6z2 + 23a-6z4 - 2a-6z6 - a-6z8 - 8a-5z-1 + 23a-5z - 35a-5z3 + 24a-5z5 - 5a-5z7 + 8a-4 - 13a-4z2 + 3a-4z4 + 3a-4z6 - a-4z8 - 4a-3z-1 + 12a-3z - 13a-3z3 + 6a-3z5 - a-3z7

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 18                 1 j = 16               2   j = 14             2 2   j = 12           3 2     j = 10         1 2 1     j = 8       2 3         j = 6     2 1           j = 4   1 4             j = 2                   j = 0 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]=	2
In[3]:=	Show[DrawMorseLink[Link[10, NonAlternating, 62]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 62]]
Out[4]=	PD[X[12, 1, 13, 2], X[16, 8, 17, 7], X[5, 1, 6, 10], X[3, 7, 4, 6], > X[9, 5, 10, 4], X[18, 14, 19, 13], X[20, 16, 11, 15], X[14, 20, 15, 19], > X[2, 11, 3, 12], X[8, 18, 9, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, -4, 5, -3, 4, 2, -10, -5, 3}, > {9, -1, 6, -8, 7, -2, 10, -6, 8, -7}]
In[6]:=	Jones[L][q]
Out[6]=	3/2 5/2 7/2 9/2 11/2 13/2 15/2 17/2 -Sqrt[q] + q - 4 q + 3 q - 4 q + 4 q - 3 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 4 6 8 10 12 18 20 22 24 26 30 q + q + 3 q + 4 q + 4 q + 4 q - 4 q - 3 q - 2 q - q + q + q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 62]][a, z]
Out[8]=	3 3 3 1 5 8 4 z 8 z 15 z 8 z 5 z 13 z 5 z -(----) + ---- - ---- + ---- - -- + --- - ---- + --- + ---- - ----- + ---- + 9 7 5 3 9 7 5 3 7 5 3 a z a z a z a z a a a a a a a 5 5 5 7 z 6 z z z > -- - ---- + -- - -- 7 5 3 5 a a a a
In[9]:=	Kauffman[Link[10, NonAlternating, 62]][a, z]
Out[9]=	2 9 14 8 1 5 8 4 z z 11 z 23 z --- + -- + -- + -- - ---- - ---- - ---- - ---- - --- - -- + ---- + ---- + 10 8 6 4 9 7 5 3 11 9 7 5 a a a a a z a z a z a z a a a a 2 2 2 2 3 3 3 3 12 z 3 z 23 z 33 z 13 z 4 z 18 z 35 z 13 z > ---- - ---- - ----- - ----- - ----- + ---- - ----- - ----- - ----- + 3 10 8 6 4 9 7 5 3 a a a a a a a a a 4 4 4 5 5 5 5 6 6 6 20 z 23 z 3 z 2 z 16 z 24 z 6 z 5 z 2 z 3 z > ----- + ----- + ---- - ---- + ----- + ----- + ---- - ---- - ---- + ---- - 8 6 4 9 7 5 3 8 6 4 a a a a a a a a a a 7 7 7 8 8 4 z 5 z z z z > ---- - ---- - -- - -- - -- 7 5 3 6 4 a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	4 4 6 -2 q 6 8 8 2 10 2 10 3 4 q + 2 q + t + -- + q t + 2 q t + 3 q t + q t + 2 q t + t 12 3 10 4 12 4 14 4 14 5 16 5 18 6 > 3 q t + q t + 2 q t + 2 q t + 2 q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n62

L10n61 L10n63