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L10a135 L10a137

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-5 + 4q-4 - 8q-3 + 13q-2 - 15q-1 + 18 - 15q + 13q2 - 8q3 + 4q4 - q5

A2 (sl(3)) Invariant:

- q-16 + q-14 + 2q-12 - 3q-10 + 3q-8 + q-6 + q-4 + 8q-2 + 3 + 8q2 + q4 + q6 + 3q8 - 3q10 + 2q12 + q14 - q16

HOMFLY-PT Polynomial:

- a-4z2 + a-2z-2 + a-2 + 2a-2z2 + 2a-2z4 - 2z-2 - 2 - 3z2 - 2z4 - z6 + a2z-2 + a2 + 2a2z2 + 2a2z4 - a4z2

Kauffman Polynomial:

- a-5z3 + a-5z5 + 3a-4z2 - 6a-4z4 + 4a-4z6 + 5a-3z3 - 11a-3z5 + 7a-3z7 + a-2z-2 - 2a-2 + 5a-2z2 - 7a-2z4 - 3a-2z6 + 6a-2z8 - 2a-1z-1 + 2a-1z + 12a-1z3 - 28a-1z5 + 13a-1z7 + 2a-1z9 + 2z-2 - 3 + 4z2 - 2z4 - 14z6 + 12z8 - 2az-1 + 2az + 12az3 - 28az5 + 13az7 + 2az9 + a2z-2 - 2a2 + 5a2z2 - 7a2z4 - 3a2z6 + 6a2z8 + 5a3z3 - 11a3z5 + 7a3z7 + 3a4z2 - 6a4z4 + 4a4z6 - a5z3 + 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                   3   j = 7                 5 1   j = 5               8 3     j = 3             8 6       j = 1           10 7         j = -1         7 10           j = -3       6 8             j = -5     3 8               j = -7   1 5                 j = -9   3                   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, 136]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 136]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[20, 7, 11, 8], X[8, 19, 9, 20], > X[18, 12, 19, 11], X[16, 13, 17, 14], X[14, 6, 15, 5], X[10, 16, 5, 15], > X[2, 9, 3, 10], X[4, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {7, -1, 3, -4, 9, -8}, > {5, -2, 6, -7, 8, -6, 10, -5, 4, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-5 4 8 13 15 2 3 4 5 18 - q + -- - -- + -- - -- - 15 q + 13 q - 8 q + 4 q - q 4 3 2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -14 2 3 3 -6 -4 8 2 4 6 8 3 - q + q + --- - --- + -- + q + q + -- + 8 q + q + q + 3 q - 12 10 8 2 q q q q 10 12 14 16 > 3 q + 2 q + q - q
In[8]:=	HOMFLYPT[Link[10, Alternating, 136]][a, z]
Out[8]=	2 2 2 -2 2 2 1 a 2 z 2 z 2 2 4 2 4 -2 + a + a - -- + ----- + -- - 3 z - -- + ---- + 2 a z - a z - 2 z + 2 2 2 2 4 2 z a z z a a 4 2 z 2 4 6 > ---- + 2 a z - z 2 a
In[9]:=	Kauffman[Link[10, Alternating, 136]][a, z]
Out[9]=	2 2 2 2 2 1 a 2 2 a 2 z 2 3 z -3 - -- - 2 a + -- + ----- + -- - --- - --- + --- + 2 a z + 4 z + ---- + 2 2 2 2 2 a z z a 4 a z a z z a 2 3 3 3 5 z 2 2 4 2 z 5 z 12 z 3 3 3 5 3 > ---- + 5 a z + 3 a z - -- + ---- + ----- + 12 a z + 5 a z - a z - 2 5 3 a a a a 4 4 5 5 5 4 6 z 7 z 2 4 4 4 z 11 z 28 z 5 > 2 z - ---- - ---- - 7 a z - 6 a z + -- - ----- - ----- - 28 a z - 4 2 5 3 a a a a a 6 6 7 7 3 5 5 5 6 4 z 3 z 2 6 4 6 7 z 13 z > 11 a z + a z - 14 z + ---- - ---- - 3 a z + 4 a z + ---- + ----- + 4 2 3 a a a a 8 9 7 3 7 8 6 z 2 8 2 z 9 > 13 a z + 7 a z + 12 z + ---- + 6 a z + ---- + 2 a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	10 1 3 1 5 3 8 6 8 -- + 10 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 7 3 3 2 5 2 5 3 7 3 7 4 > --- + 7 q t + 8 q t + 6 q t + 8 q t + 3 q t + 5 q t + q t + q t 9 4 11 5 > 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a136

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