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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X12,6,13,5 X8493 X2,14,3,13 X14,7,15,8 X18,10,19,9 X16,12,17,11 X20,18,11,17 X4,15,1,16 X10,20,5,19

Gauss Code:

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

Jones Polynomial:

- q-2 + 4q-1 - 7 + 13q - 14q2 + 17q3 - 15q4 + 12q5 - 8q6 + 4q7 - q8

A2 (sl(3)) Invariant:

- q-6 + 2q-4 + 1 + 7q2 + 2q4 + 8q6 + 4q8 + 2q10 + 3q12 - 3q14 + 3q16 - q18 - q20 + 2q22 - q24

HOMFLY-PT Polynomial:

- a-6z2 - a-6z4 + a-4z-2 + a-4z2 + 2a-4z4 + a-4z6 - 2a-2z-2 - a-2 + a-2z2 + 2a-2z4 + a-2z6 + z-2 + 1 - z2 - z4

Kauffman Polynomial:

- a-9z3 + a-9z5 + 2a-8z2 - 6a-8z4 + 4a-8z6 + 6a-7z3 - 12a-7z5 + 7a-7z7 + 2a-6z2 - 3a-6z4 - 5a-6z6 + 6a-6z8 + 11a-5z3 - 24a-5z5 + 11a-5z7 + 2a-5z9 - a-4z-2 + a-4 + 2a-4z2 - a-4z4 - 13a-4z6 + 11a-4z8 + 2a-3z-1 - a-3z + 7a-3z3 - 20a-3z5 + 10a-3z7 + 2a-3z9 - 2a-2z-2 + a-2 + 6a-2z2 - 11a-2z4 + 5a-2z8 + 2a-1z-1 - a-1z + 2a-1z3 - 8a-1z5 + 6a-1z7 - z-2 + 1 + 4z2 - 7z4 + 4z6 - az3 + az5

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 17                     1 j = 15                   3   j = 13                 5 1   j = 11               7 3     j = 9             8 5       j = 7           9 7         j = 5         7 10           j = 3       6 7             j = 1     3 9               j = -1   1 4                 j = -3   3                   j = -5 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, 127]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 127]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 6, 13, 5], X[8, 4, 9, 3], X[2, 14, 3, 13], > X[14, 7, 15, 8], X[18, 10, 19, 9], X[16, 12, 17, 11], X[20, 18, 11, 17], > X[4, 15, 1, 16], X[10, 20, 5, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -9}, {2, -1, 5, -3, 6, -10}, > {7, -2, 4, -5, 9, -7, 8, -6, 10, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-2 4 2 3 4 5 6 7 8 -7 - q + - + 13 q - 14 q + 17 q - 15 q + 12 q - 8 q + 4 q - q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 2 2 4 6 8 10 12 14 16 1 - q + -- + 7 q + 2 q + 8 q + 4 q + 2 q + 3 q - 3 q + 3 q - 4 q 18 20 22 24 > q - q + 2 q - q
In[8]:=	HOMFLYPT[Link[10, Alternating, 127]][a, z]
Out[8]=	2 2 2 4 4 4 -2 -2 1 2 2 z z z 4 z 2 z 2 z 1 - a + z + ----- - ----- - z - -- + -- + -- - z - -- + ---- + ---- + 4 2 2 2 6 4 2 6 4 2 a z a z a a a a a a 6 6 z z > -- + -- 4 2 a a
In[9]:=	Kauffman[Link[10, Alternating, 127]][a, z]
Out[9]=	2 -4 -2 -2 1 2 2 2 z z 2 2 z 1 + a + a - z - ----- - ----- + ---- + --- - -- - - + 4 z + ---- + 4 2 2 2 3 a z 3 a 8 a z a z a z a a 2 2 2 3 3 3 3 3 4 2 z 2 z 6 z z 6 z 11 z 7 z 2 z 3 4 6 z > ---- + ---- + ---- - -- + ---- + ----- + ---- + ---- - a z - 7 z - ---- - 6 4 2 9 7 5 3 a 8 a a a a a a a a 4 4 4 5 5 5 5 5 3 z z 11 z z 12 z 24 z 20 z 8 z 5 6 > ---- - -- - ----- + -- - ----- - ----- - ----- - ---- + a z + 4 z + 6 4 2 9 7 5 3 a a a a a a a a 6 6 6 7 7 7 7 8 8 8 4 z 5 z 13 z 7 z 11 z 10 z 6 z 6 z 11 z 5 z > ---- - ---- - ----- + ---- + ----- + ----- + ---- + ---- + ----- + ---- + 8 6 4 7 5 3 a 6 4 2 a a a a a a a a a 9 9 2 z 2 z > ---- + ---- 5 3 a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 3 1 4 3 q 3 5 5 2 9 q + 6 q + ----- + ----- + ---- + --- + --- + 7 q t + 7 q t + 10 q t + 5 3 3 2 2 q t t q t q t q t 7 2 7 3 9 3 9 4 11 4 11 5 13 5 > 9 q t + 7 q t + 8 q t + 5 q t + 7 q t + 3 q t + 5 q t + 13 6 15 6 17 7 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a127

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