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L10n65 L10n67

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X3,11,4,10 X16,12,17,11 X18,13,19,14 X20,18,9,17 X12,19,13,20 X15,8,16,5 X7,14,8,15 X2536 X9,1,10,4

Gauss Code:

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

Jones Polynomial:

q-6 - q-5 + 2q-4 - q-3 + q-2 + 1 + q - q2 + q3

A2 (sl(3)) Invariant:

q-20 + 2q-18 + 2q-16 + 3q-14 + 2q-12 + 2q-10 + q-8 + q-4 + q-2 + 3 + 3q2 + 2q4 + 2q6 + q8 + q10

HOMFLY-PT Polynomial:

a-2z-2 + 2a-2 + a-2z2 - 3z-2 - 6 - 5z2 - z4 + 4a2z-2 + 8a2 + 5a2z2 + a2z4 - 3a4z-2 - 5a4 - 2a4z2 + a6z-2 + a6

Kauffman Polynomial:

a-2z-2 - 4a-2 + 6a-2z2 - 5a-2z4 + a-2z6 - a-1z-1 + a-1z + 4a-1z3 - 5a-1z5 + a-1z7 + 3z-2 - 14 + 25z2 - 14z4 + 2z6 - az-1 + az + 6az3 - 6az5 + az7 + 4a2z-2 - 21a2 + 35a2z2 - 20a2z4 + 3a2z6 - a3z-1 + a3z + 4a3z3 - 5a3z5 + a3z7 + 3a4z-2 - 14a4 + 23a4z2 - 16a4z4 + 3a4z6 - a5z-1 + a5z + 2a5z3 - 4a5z5 + a5z7 + a6z-2 - 4a6 + 7a6z2 - 5a6z4 + a6z6

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                     1 j = 5                       j = 3                 1 1   j = 1             3 1       j = -1           1 3 1       j = -3         1 2 2         j = -5       1 1             j = -7     1 1 1             j = -9   1 2                 j = -11                       j = -13 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[10, NonAlternating, 66]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 66]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 11, 4, 10], X[16, 12, 17, 11], X[18, 13, 19, 14], > X[20, 18, 9, 17], X[12, 19, 13, 20], X[15, 8, 16, 5], X[7, 14, 8, 15], > X[2, 5, 3, 6], X[9, 1, 10, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, -2, 10}, {9, -1, -8, 7}, > {-10, 2, 3, -6, 4, 8, -7, -3, 5, -4, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-6 -5 2 -3 -2 2 3 1 + q - q + -- - q + q + q - q + q 4 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 2 2 3 2 2 -8 -4 -2 2 4 6 3 + q + --- + --- + --- + --- + --- + q + q + q + 3 q + 2 q + 2 q + 18 16 14 12 10 q q q q q 8 10 > q + q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 66]][a, z]
Out[8]=	2 4 6 2 2 2 4 6 3 1 4 a 3 a a 2 z -6 + -- + 8 a - 5 a + a - -- + ----- + ---- - ---- + -- - 5 z + -- + 2 2 2 2 2 2 2 2 a z a z z z z a 2 2 4 2 4 2 4 > 5 a z - 2 a z - z + a z
In[9]:=	Kauffman[Link[10, NonAlternating, 66]][a, z]
Out[9]=	2 4 6 4 2 4 6 3 1 4 a 3 a a 1 a -14 - -- - 21 a - 14 a - 4 a + -- + ----- + ---- + ---- + -- - --- - - - 2 2 2 2 2 2 2 a z z a z a z z z z 3 5 2 a a z 3 5 2 6 z 2 2 4 2 > -- - -- + - + a z + a z + a z + 25 z + ---- + 35 a z + 23 a z + z z a 2 a 3 4 6 2 4 z 3 3 3 5 3 4 5 z 2 4 > 7 a z + ---- + 6 a z + 4 a z + 2 a z - 14 z - ---- - 20 a z - a 2 a 5 6 4 4 6 4 5 z 5 3 5 5 5 6 z > 16 a z - 5 a z - ---- - 6 a z - 5 a z - 4 a z + 2 z + -- + a 2 a 7 2 6 4 6 6 6 z 7 3 7 5 7 > 3 a z + 3 a z + a z + -- + a z + a z + a z a
In[10]:=	Kh[L][q, t]
Out[10]=	2 3 1 1 2 1 1 1 1 1 -- + - + 3 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ----- + 3 q 13 6 9 5 9 4 7 4 7 3 5 3 7 2 5 2 q q t q t q t q t q t q t q t q t 1 2 1 t 3 2 3 3 7 4 > ----- + ---- + --- + - + q t + q t + q t + q t 3 2 3 q t q q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n66

L10n65 L10n67