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Knotscape

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DrawMorseLink

PD Presentation:

X10,1,11,2 X16,11,17,12 X8,9,1,10 X20,17,9,18 X12,4,13,3 X14,8,15,7 X6,14,7,13 X18,6,19,5 X4,20,5,19 X2,16,3,15

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 4q-5/2 - 8q-3/2 + 12q-1/2 - 15q1/2 + 15q3/2 - 15q5/2 + 11q7/2 - 7q9/2 + 3q11/2 - q13/2

A2 (sl(3)) Invariant:

q-10 - 2q-8 + 2q-6 - q-4 - 2q-2 + 3 - 3q2 + 4q4 + 2q8 + 3q10 - 2q12 + 3q14 + q20

HOMFLY-PT Polynomial:

- a-5z-1 - 2a-5z - a-5z3 + a-3z-1 + 6a-3z + 6a-3z3 + 2a-3z5 - 3a-1z - 6a-1z3 - 4a-1z5 - a-1z7 + az + 2az3 + az5

Kauffman Polynomial:

- a-7z + 2a-7z3 - a-7z5 - 2a-6z2 + 5a-6z4 - 3a-6z6 - a-5z-1 + 4a-5z - 6a-5z3 + 8a-5z5 - 5a-5z7 + a-4 - 2a-4z2 + 5a-4z6 - 5a-4z8 - a-3z-1 + 9a-3z - 23a-3z3 + 25a-3z5 - 9a-3z7 - 2a-3z9 - a-2z2 - 6a-2z4 + 17a-2z6 - 11a-2z8 + 6a-1z - 22a-1z3 + 29a-1z5 - 11a-1z7 - 2a-1z9 - 3z2 + 5z4 + 5z6 - 6z8 + 2az - 6az3 + 12az5 - 7az7 - 2a2z2 + 6a2z4 - 4a2z6 + a3z3 - a3z5

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 = 14                     1 j = 12                   2   j = 10                 5 1   j = 8               7 3     j = 6             8 4       j = 4           7 7         j = 2         8 8           j = 0       5 8             j = -2     3 7               j = -4   1 5                 j = -6   3                   j = -8 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, 104]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 104]]
Out[4]=	PD[X[10, 1, 11, 2], X[16, 11, 17, 12], X[8, 9, 1, 10], X[20, 17, 9, 18], > X[12, 4, 13, 3], X[14, 8, 15, 7], X[6, 14, 7, 13], X[18, 6, 19, 5], > X[4, 20, 5, 19], X[2, 16, 3, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 5, -9, 8, -7, 6, -3}, > {3, -1, 2, -5, 7, -6, 10, -2, 4, -8, 9, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 4 8 12 3/2 5/2 7/2 -q + ---- - ---- + ------- - 15 Sqrt[q] + 15 q - 15 q + 11 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 13/2 > 7 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 2 2 -4 2 2 4 8 10 12 14 20 3 + q - -- + -- - q - -- - 3 q + 4 q + 2 q + 3 q - 2 q + 3 q + q 8 6 2 q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 104]][a, z]
Out[8]=	3 3 3 5 1 1 2 z 6 z 3 z z 6 z 6 z 3 2 z -(----) + ---- - --- + --- - --- + a z - -- + ---- - ---- + 2 a z + ---- - 5 3 5 3 a 5 3 a 3 a z a z a a a a a 5 7 4 z 5 z > ---- + a z - -- a a
In[9]:=	Kauffman[Link[10, Alternating, 104]][a, z]
Out[9]=	2 2 2 -4 1 1 z 4 z 9 z 6 z 2 2 z 2 z z a - ---- - ---- - -- + --- + --- + --- + 2 a z - 3 z - ---- - ---- - -- - 5 3 7 5 3 a 6 4 2 a z a z a a a a a a 3 3 3 3 4 2 2 2 z 6 z 23 z 22 z 3 3 3 4 5 z > 2 a z + ---- - ---- - ----- - ----- - 6 a z + a z + 5 z + ---- - 7 5 3 a 6 a a a a 4 5 5 5 5 6 z 2 4 z 8 z 25 z 29 z 5 3 5 6 > ---- + 6 a z - -- + ---- + ----- + ----- + 12 a z - a z + 5 z - 2 7 5 3 a a a a a 6 6 6 7 7 7 3 z 5 z 17 z 2 6 5 z 9 z 11 z 7 8 > ---- + ---- + ----- - 4 a z - ---- - ---- - ----- - 7 a z - 6 z - 6 4 2 5 3 a a a a a a 8 8 9 9 5 z 11 z 2 z 2 z > ---- - ----- - ---- - ---- 4 2 3 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 3 1 5 3 5 7 2 4 8 + 8 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 8 q t + 7 q t + 8 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t 4 2 6 2 6 3 8 3 8 4 10 4 10 5 > 7 q t + 8 q t + 4 q t + 7 q t + 3 q t + 5 q t + q t + 12 5 14 6 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a104

L10a103 L10a105