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L10a138 L10a140

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X18,12,19,11 X14,8,15,7 X8,14,9,13 X20,15,13,16 X16,19,17,20 X12,18,5,17 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-6 - 3q-5 + 6q-4 - 8q-3 + 10q-2 - 10q-1 + 11 - 7q + 5q2 - 2q3 + q4

A2 (sl(3)) Invariant:

q-20 + q-18 - 2q-16 + q-14 - 2q-10 + 3q-8 + q-6 + 4q-4 + 5q-2 + 4 + 6q2 + 2q6 + 2q8 - q10 + q12 + q14

HOMFLY-PT Polynomial:

a-4 + a-2z-2 - 2a-2z2 - 2z-2 - 3 - 2z2 + z4 + a2z-2 + 3a2 + 3a2z2 + 2a2z4 - 2a4 - 3a4z2 + a6

Kauffman Polynomial:

a-4 - 2a-4z2 + a-4z4 - 2a-3z3 + 2a-3z5 + a-2z-2 - 3a-2 + 3a-2z2 - 3a-2z4 + 3a-2z6 - 2a-1z-1 + 6a-1z - 5a-1z3 + 3a-1z7 + 2z-2 - 7 + 10z2 - 3z4 - 2z6 + 3z8 - 2az-1 + 2az + 11az3 - 17az5 + 6az7 + az9 + a2z-2 - 4a2 + 8a2z2 - a2z4 - 11a2z6 + 6a2z8 - 6a3z + 21a3z3 - 24a3z5 + 6a3z7 + a3z9 - a4 + 6a4z2 - 5a4z4 - 5a4z6 + 3a4z8 - 2a5z + 7a5z3 - 9a5z5 + 3a5z7 - a6 + 3a6z2 - 3a6z4 + 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 = 9                     1 j = 7                   2 1 j = 5                 3     j = 3               4 2     j = 1             7 3       j = -1           5 6         j = -3         5 5           j = -5       3 5             j = -7     3 5               j = -9   1 4                 j = -11   2                   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, Alternating, 139]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 139]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[18, 12, 19, 11], X[14, 8, 15, 7], > X[8, 14, 9, 13], X[20, 15, 13, 16], X[16, 19, 17, 20], X[12, 18, 5, 17], > X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 4, -5, 10, -2, 3, -8}, > {5, -4, 6, -7, 8, -3, 7, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-6 3 6 8 10 10 2 3 4 11 + q - -- + -- - -- + -- - -- - 7 q + 5 q - 2 q + q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -18 2 -14 2 3 -6 4 5 2 6 8 4 + q + q - --- + q - --- + -- + q + -- + -- + 6 q + 2 q + 2 q - 16 10 8 4 2 q q q q q 10 12 14 > q + q + q
In[8]:=	HOMFLYPT[Link[10, Alternating, 139]][a, z]
Out[8]=	2 2 -4 2 4 6 2 1 a 2 2 z 2 2 -3 + a + 3 a - 2 a + a - -- + ----- + -- - 2 z - ---- + 3 a z - 2 2 2 2 2 z a z z a 4 2 4 2 4 > 3 a z + z + 2 a z
In[9]:=	Kauffman[Link[10, Alternating, 139]][a, z]
Out[9]=	2 -4 3 2 4 6 2 1 a 2 2 a 6 z -7 + a - -- - 4 a - a - a + -- + ----- + -- - --- - --- + --- + 2 a z - 2 2 2 2 2 a z z a a z a z z 2 2 3 5 2 2 z 3 z 2 2 4 2 6 2 > 6 a z - 2 a z + 10 z - ---- + ---- + 8 a z + 6 a z + 3 a z - 4 2 a a 3 3 4 4 2 z 5 z 3 3 3 5 3 4 z 3 z 2 4 > ---- - ---- + 11 a z + 21 a z + 7 a z - 3 z + -- - ---- - a z - 3 a 4 2 a a a 5 6 4 4 6 4 2 z 5 3 5 5 5 6 3 z > 5 a z - 3 a z + ---- - 17 a z - 24 a z - 9 a z - 2 z + ---- - 3 2 a a 7 2 6 4 6 6 6 3 z 7 3 7 5 7 8 > 11 a z - 5 a z + a z + ---- + 6 a z + 6 a z + 3 a z + 3 z + a 2 8 4 8 9 3 9 > 6 a z + 3 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	6 1 2 1 4 3 5 3 5 - + 7 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 5 5 5 3 3 2 5 2 7 3 7 4 > ----- + ---- + --- + 3 q t + 4 q t + 2 q t + 3 q t + 2 q t + q t + 3 2 3 q t q t q t 9 4 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a139

L10a138 L10a140