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L10a147 L10a149

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-3 - 3q-2 + 5q-1 - 6 + 10q - 9q2 + 10q3 - 7q4 + 5q5 - 3q6 + q7

A2 (sl(3)) Invariant:

q-8 - q-6 + q-4 + 2 + 5q2 + 3q4 + 8q6 + 3q8 + 4q10 + q12 - q14 + q16 - q18 + q20

HOMFLY-PT Polynomial:

a-4z-2 + a-4 + 4a-4z2 + 4a-4z4 + a-4z6 - 2a-2z-2 - 3a-2 - 9a-2z2 - 12a-2z4 - 6a-2z6 - a-2z8 + z-2 + 2 + 4z2 + 4z4 + z6

Kauffman Polynomial:

- a-8z2 + a-8z4 - 4a-7z3 + 3a-7z5 + a-6z2 - 5a-6z4 + 4a-6z6 + 2a-5z3 - 5a-5z5 + 4a-5z7 - a-4z-2 + 3a-4 - 10a-4z2 + 15a-4z4 - 10a-4z6 + 4a-4z8 + 2a-3z-1 - 3a-3z + 7a-3z3 - 3a-3z5 - 2a-3z7 + 2a-3z9 - 2a-2z-2 + 5a-2 - 22a-2z2 + 41a-2z4 - 30a-2z6 + 8a-2z8 + 2a-1z-1 - 3a-1z + 7a-1z3 - 5a-1z5 - 3a-1z7 + 2a-1z9 - z-2 + 3 - 9z2 + 17z4 - 15z6 + 4z8 + 6az3 - 10az5 + 3az7 + a2z2 - 3a2z4 + 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 j = 15                     1 j = 13                   2   j = 11                 3 1   j = 9               5 3     j = 7             5 2       j = 5           4 5         j = 3         6 5           j = 1       3 7             j = -1     2 3               j = -3   1 3                 j = -5   2                   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[10, Alternating, 148]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 148]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 4, 15, 3], X[16, 5, 17, 6], X[12, 15, 5, 16], > X[8, 20, 9, 19], X[18, 8, 19, 7], X[20, 10, 13, 9], X[10, 14, 11, 13], > X[2, 11, 3, 12], X[4, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {3, -1, 6, -5, 7, -8, 9, -4}, > {8, -2, 4, -3, 10, -6, 5, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-3 3 5 2 3 4 5 6 7 -6 + q - -- + - + 10 q - 9 q + 10 q - 7 q + 5 q - 3 q + q 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 -6 -4 2 4 6 8 10 12 14 16 2 + q - q + q + 5 q + 3 q + 8 q + 3 q + 4 q + q - q + q - 18 20 > q + q
In[8]:=	HOMFLYPT[Link[10, Alternating, 148]][a, z]
Out[8]=	2 2 4 4 -4 3 -2 1 2 2 4 z 9 z 4 4 z 12 z 2 + a - -- + z + ----- - ----- + 4 z + ---- - ---- + 4 z + ---- - ----- + 2 4 2 2 2 4 2 4 2 a a z a z a a a a 6 6 8 6 z 6 z z > z + -- - ---- - -- 4 2 2 a a a
In[9]:=	Kauffman[Link[10, Alternating, 148]][a, z]
Out[9]=	2 2 3 5 -2 1 2 2 2 3 z 3 z 2 z z 3 + -- + -- - z - ----- - ----- + ---- + --- - --- - --- - 9 z - -- + -- - 4 2 4 2 2 2 3 a z 3 a 8 6 a a a z a z a z a a a 2 2 3 3 3 3 4 10 z 22 z 2 2 4 z 2 z 7 z 7 z 3 4 z > ----- - ----- + a z - ---- + ---- + ---- + ---- + 6 a z + 17 z + -- - 4 2 7 5 3 a 8 a a a a a a 4 4 4 5 5 5 5 5 z 15 z 41 z 2 4 3 z 5 z 3 z 5 z 5 > ---- + ----- + ----- - 3 a z + ---- - ---- - ---- - ---- - 10 a z - 6 4 2 7 5 3 a a a a a a a 6 6 6 7 7 7 6 4 z 10 z 30 z 2 6 4 z 2 z 3 z 7 8 > 15 z + ---- - ----- - ----- + a z + ---- - ---- - ---- + 3 a z + 4 z + 6 4 2 5 3 a a a a a a 8 8 9 9 4 z 8 z 2 z 2 z > ---- + ---- + ---- + ---- 4 2 3 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 2 1 3 2 3 3 q 3 7 q + 6 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 5 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 > 4 q t + 5 q t + 5 q t + 2 q t + 5 q t + 3 q t + 3 q t + 11 5 13 5 15 6 > q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a148

L10a147 L10a149