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L10a109 L10a111

Knotscape

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DrawMorseLink

PD Presentation:

X12,1,13,2 X8493 X18,14,19,13 X20,16,11,15 X14,20,15,19 X6,17,7,18 X16,7,17,8 X10,6,1,5 X4,10,5,9 X2,11,3,12

Gauss Code:

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

Jones Polynomial:

- q-5/2 + q-3/2 - 5q-1/2 + 6q1/2 - 9q3/2 + 10q5/2 - 9q7/2 + 9q9/2 - 6q11/2 + 3q13/2 - q15/2

A2 (sl(3)) Invariant:

q-8 + 2q-6 + 3q-4 + 6q-2 + 3 + 3q2 - 4q6 - q8 - 5q10 - q14 + 2q18 - q20 + q22

HOMFLY-PT Polynomial:

- a-5z-1 - 3a-5z - 3a-5z3 - a-5z5 + 5a-3z-1 + 11a-3z + 10a-3z3 + 5a-3z5 + a-3z7 - 8a-1z-1 - 14a-1z - 9a-1z3 - 2a-1z5 + 4az-1 + 4az + az3

Kauffman Polynomial:

- a-9z3 - 3a-8z4 - a-7z + 4a-7z3 - 6a-7z5 + 2a-6 - 9a-6z2 + 15a-6z4 - 9a-6z6 - a-5z-1 - 4a-5z3 + 14a-5z5 - 8a-5z7 + 9a-4 - 23a-4z2 + 22a-4z4 - 4a-4z8 - 5a-3z-1 + 9a-3z - 20a-3z3 + 26a-3z5 - 7a-3z7 - a-3z9 + 14a-2 - 28a-2z2 + 9a-2z4 + 11a-2z6 - 5a-2z8 - 8a-1z-1 + 20a-1z - 24a-1z3 + 12a-1z5 - a-1z9 + 8 - 14z2 + 5z4 + 2z6 - z8 - 4az-1 + 12az - 13az3 + 6az5 - az7

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 = 16                     1 j = 14                   2   j = 12                 4 1   j = 10               5 2     j = 8             4 4       j = 6           6 5         j = 4         3 4           j = 2       4 7             j = 0     1 2               j = -2     4                 j = -4 1 1                   j = -6 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]=	2
In[3]:=	Show[DrawMorseLink[Link[10, Alternating, 110]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 110]]
Out[4]=	PD[X[12, 1, 13, 2], X[8, 4, 9, 3], X[18, 14, 19, 13], X[20, 16, 11, 15], > X[14, 20, 15, 19], X[6, 17, 7, 18], X[16, 7, 17, 8], X[10, 6, 1, 5], > X[4, 10, 5, 9], X[2, 11, 3, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -9, 8, -6, 7, -2, 9, -8}, > {10, -1, 3, -5, 4, -7, 6, -3, 5, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) -(3/2) 5 3/2 5/2 7/2 9/2 -q + q - ------- + 6 Sqrt[q] - 9 q + 10 q - 9 q + 9 q - Sqrt[q] 11/2 13/2 15/2 > 6 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 2 3 6 2 6 8 10 14 18 20 22 3 + q + -- + -- + -- + 3 q - 4 q - q - 5 q - q + 2 q - q + q 6 4 2 q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 110]][a, z]
Out[8]=	3 3 3 1 5 8 4 a 3 z 11 z 14 z 3 z 10 z 9 z -(----) + ---- - --- + --- - --- + ---- - ---- + 4 a z - ---- + ----- - ---- + 5 3 a z z 5 3 a 5 3 a a z a z a a a a 5 5 5 7 3 z 5 z 2 z z > a z - -- + ---- - ---- + -- 5 3 a 3 a a a
In[9]:=	Kauffman[Link[10, Alternating, 110]][a, z]
Out[9]=	2 9 14 1 5 8 4 a z 9 z 20 z 2 8 + -- + -- + -- - ---- - ---- - --- - --- - -- + --- + ---- + 12 a z - 14 z - 6 4 2 5 3 a z z 7 3 a a a a a z a z a a 2 2 2 3 3 3 3 3 9 z 23 z 28 z z 4 z 4 z 20 z 24 z 3 4 > ---- - ----- - ----- - -- + ---- - ---- - ----- - ----- - 13 a z + 5 z - 6 4 2 9 7 5 3 a a a a a a a a 4 4 4 4 5 5 5 5 3 z 15 z 22 z 9 z 6 z 14 z 26 z 12 z 5 > ---- + ----- + ----- + ---- - ---- + ----- + ----- + ----- + 6 a z + 8 6 4 2 7 5 3 a a a a a a a a 6 6 7 7 8 8 9 9 6 9 z 11 z 8 z 7 z 7 8 4 z 5 z z z > 2 z - ---- + ----- - ---- - ---- - a z - z - ---- - ---- - -- - -- 6 2 5 3 4 2 3 a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 1 1 -2 4 2 4 q 4 7 q + 3 q + ----- + ----- + ----- + t + ----- + - + ---- + 4 q t + 6 4 4 4 4 3 2 2 t t q t q t q t q t 6 6 2 8 2 8 3 10 3 10 4 12 4 > 6 q t + 5 q t + 4 q t + 4 q t + 5 q t + 2 q t + 4 q t + 12 5 14 5 16 6 > q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a110

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