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L10a139 L10a141

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-5 + 3q-4 - 5q-3 + 8q-2 - 9q-1 + 12 - 9q + 8q2 - 5q3 + 3q4 - q5

A2 (sl(3)) Invariant:

- q-14 + q-12 - q-10 + 2q-8 + 2q-6 + 2q-4 + 7q-2 + 3 + 7q2 + 2q4 + 2q6 + 2q8 - q10 + q12 - q14

HOMFLY-PT Polynomial:

a-2z-2 - 4a-2z2 - 4a-2z4 - a-2z6 - 2z-2 + 8z2 + 12z4 + 6z6 + z8 + a2z-2 - 4a2z2 - 4a2z4 - a2z6

Kauffman Polynomial:

- 2a-5z3 + a-5z5 + 2a-4z2 - 7a-4z4 + 3a-4z6 + 4a-3z3 - 9a-3z5 + 4a-3z7 + a-2z-2 - 8a-2z2 + 14a-2z4 - 11a-2z6 + 4a-2z8 - 2a-1z-1 + 6a-1z3 - 2a-1z5 - 2a-1z7 + 2a-1z9 + 2z-2 + 1 - 20z2 + 42z4 - 28z6 + 8z8 - 2az-1 + 6az3 - 2az5 - 2az7 + 2az9 + a2z-2 - 8a2z2 + 14a2z4 - 11a2z6 + 4a2z8 + 4a3z3 - 9a3z5 + 4a3z7 + 2a4z2 - 7a4z4 + 3a4z6 - 2a5z3 + a5z5

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 11                     1 j = 9                   2   j = 7                 3 1   j = 5               5 2     j = 3             4 3       j = 1           8 5         j = -1         5 8           j = -3       3 4             j = -5     2 5               j = -7   1 3                 j = -9   2                   j = -11 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, 140]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 140]]
Out[4]=	PD[X[6, 1, 7, 2], X[2, 16, 3, 15], X[10, 4, 11, 3], X[14, 6, 15, 5], > X[20, 12, 13, 11], X[12, 14, 5, 13], X[4, 19, 1, 20], X[8, 17, 9, 18], > X[16, 7, 17, 8], X[18, 9, 19, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -7}, {4, -1, 9, -8, 10, -3, 5, -6}, > {6, -4, 2, -9, 8, -10, 7, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-5 3 5 8 9 2 3 4 5 12 - q + -- - -- + -- - - - 9 q + 8 q - 5 q + 3 q - q 4 3 2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -12 -10 2 2 2 7 2 4 6 8 10 3 - q + q - q + -- + -- + -- + -- + 7 q + 2 q + 2 q + 2 q - q + 8 6 4 2 q q q q 12 14 > q - q
In[8]:=	HOMFLYPT[Link[10, Alternating, 140]][a, z]
Out[8]=	2 2 4 6 -2 1 a 2 4 z 2 2 4 4 z 2 4 6 z -- + ----- + -- + 8 z - ---- - 4 a z + 12 z - ---- - 4 a z + 6 z - -- - 2 2 2 2 2 2 2 z a z z a a a 2 6 8 > a z + z
In[9]:=	Kauffman[Link[10, Alternating, 140]][a, z]
Out[9]=	2 2 2 2 1 a 2 2 a 2 2 z 8 z 2 2 4 2 1 + -- + ----- + -- - --- - --- - 20 z + ---- - ---- - 8 a z + 2 a z - 2 2 2 2 a z z 4 2 z a z z a a 3 3 3 4 4 2 z 4 z 6 z 3 3 3 5 3 4 7 z 14 z > ---- + ---- + ---- + 6 a z + 4 a z - 2 a z + 42 z - ---- + ----- + 5 3 a 4 2 a a a a 5 5 5 2 4 4 4 z 9 z 2 z 5 3 5 5 5 6 > 14 a z - 7 a z + -- - ---- - ---- - 2 a z - 9 a z + a z - 28 z + 5 3 a a a 6 6 7 7 3 z 11 z 2 6 4 6 4 z 2 z 7 3 7 8 > ---- - ----- - 11 a z + 3 a z + ---- - ---- - 2 a z + 4 a z + 8 z + 4 2 3 a a a a 8 9 4 z 2 8 2 z 9 > ---- + 4 a z + ---- + 2 a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	8 1 2 1 3 2 5 3 4 5 - + 8 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 7 4 9 4 > 5 q t + 4 q t + 3 q t + 5 q t + 2 q t + 3 q t + q t + 2 q t + 11 5 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a140

L10a139 L10a141