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L10a127 L10a129

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-6 - 2q-5 + 6q-4 - 8q-3 + 12q-2 - 12q-1 + 13 - 10q + 7q2 - 4q3 + q4

A2 (sl(3)) Invariant:

q-20 + 2q-18 + q-16 + 4q-14 + 5q-12 + 2q-10 + 6q-8 + 2q-6 + 2q-4 + 3q-2 - 1 + 3q2 - 3q4 + q8 - 2q10 + q12

HOMFLY-PT Polynomial:

a-2z2 + a-2z4 - 1 - 4z2 - 3z4 - z6 + a2z-2 + 5a2 + 7a2z2 + 3a2z4 - 2a4z-2 - 5a4 - 3a4z2 + a6z-2 + a6

Kauffman Polynomial:

a-4z4 - 3a-3z3 + 4a-3z5 + 3a-2z2 - 8a-2z4 + 7a-2z6 + a-1z - a-1z3 - 6a-1z5 + 7a-1z7 - 2 + 10z2 - 16z4 + 4z6 + 4z8 + 3az + 4az3 - 19az5 + 10az7 + az9 + a2z-2 - 8a2 + 19a2z2 - 16a2z4 - 4a2z6 + 6a2z8 - 2a3z-1 + 5a3z + 4a3z3 - 14a3z5 + 5a3z7 + a3z9 + 2a4z-2 - 9a4 + 18a4z2 - 13a4z4 + 2a4z8 - 2a5z-1 + 3a5z + 2a5z3 - 5a5z5 + 2a5z7 + a6z-2 - 4a6 + 6a6z2 - 4a6z4 + 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                   3   j = 5                 4 1   j = 3               6 3     j = 1             7 4       j = -1           6 7         j = -3         6 6           j = -5       4 8             j = -7     2 4               j = -9   1 5                 j = -11   1                   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, 128]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 128]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[20, 14, 11, 13], X[18, 16, 19, 15], > X[14, 8, 15, 7], X[10, 20, 5, 19], X[8, 17, 9, 18], X[16, 9, 17, 10], > X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 5, -7, 8, -6}, > {10, -2, 3, -5, 4, -8, 7, -4, 6, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-6 2 6 8 12 12 2 3 4 13 + q - -- + -- - -- + -- - -- - 10 q + 7 q - 4 q + q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 2 -16 4 5 2 6 2 2 3 2 4 -1 + q + --- + q + --- + --- + --- + -- + -- + -- + -- + 3 q - 3 q + 18 14 12 10 8 6 4 2 q q q q q q q q 8 10 12 > q - 2 q + q
In[8]:=	HOMFLYPT[Link[10, Alternating, 128]][a, z]
Out[8]=	2 4 6 2 2 4 6 a 2 a a 2 z 2 2 4 2 4 -1 + 5 a - 5 a + a + -- - ---- + -- - 4 z + -- + 7 a z - 3 a z - 3 z + 2 2 2 2 z z z a 4 z 2 4 6 > -- + 3 a z - z 2 a
In[9]:=	Kauffman[Link[10, Alternating, 128]][a, z]
Out[9]=	2 4 6 3 5 2 4 6 a 2 a a 2 a 2 a z 3 -2 - 8 a - 9 a - 4 a + -- + ---- + -- - ---- - ---- + - + 3 a z + 5 a z + 2 2 2 z z a z z z 2 3 3 5 2 3 z 2 2 4 2 6 2 3 z z > 3 a z + 10 z + ---- + 19 a z + 18 a z + 6 a z - ---- - -- + 2 3 a a a 4 4 3 3 3 5 3 4 z 8 z 2 4 4 4 > 4 a z + 4 a z + 2 a z - 16 z + -- - ---- - 16 a z - 13 a z - 4 2 a a 5 5 6 6 4 4 z 6 z 5 3 5 5 5 6 7 z > 4 a z + ---- - ---- - 19 a z - 14 a z - 5 a z + 4 z + ---- - 3 a 2 a a 7 2 6 6 6 7 z 7 3 7 5 7 8 2 8 > 4 a z + a z + ---- + 10 a z + 5 a z + 2 a z + 4 z + 6 a z + a 4 8 9 3 9 > 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	7 1 1 1 5 2 4 4 8 - + 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 6 6 6 3 3 2 5 2 5 3 7 3 > ----- + ---- + --- + 4 q t + 6 q t + 3 q t + 4 q t + q t + 3 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 L10a128

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