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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-2 + 4q-1 - 7 + 11q - 12q2 + 14q3 - 12q4 + 10q5 - 5q6 + 3q7 - q8

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

a-6z-2 - 2a-6z2 - a-6z4 - 2a-4z-2 - a-4 + 3a-4z2 + 3a-4z4 + a-4z6 + a-2z-2 + 2a-2z4 + a-2z6 + 1 - z2 - z4

Kauffman Polynomial:

- 2a-9z3 + a-9z5 + 4a-8z2 - 7a-8z4 + 3a-8z6 + 3a-7z3 - 7a-7z5 + 4a-7z7 + a-6z-2 - 3a-6 + 3a-6z2 + 3a-6z4 - 6a-6z6 + 4a-6z8 - 2a-5z-1 + 4a-5z + 3a-5z3 - 6a-5z5 + 2a-5z7 + 2a-5z9 + 2a-4z-2 - 4a-4 - 5a-4z2 + 18a-4z4 - 19a-4z6 + 9a-4z8 - 2a-3z-1 + 4a-3z + a-3z3 - 9a-3z5 + 4a-3z7 + 2a-3z9 + a-2z-2 - a-2 - 2a-2z2 + a-2z4 - 6a-2z6 + 5a-2z8 + 2a-1z3 - 10a-1z5 + 6a-1z7 + 1 + 2z2 - 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                   2   j = 13                 3 1   j = 11               7 2     j = 9             7 5       j = 7           7 5         j = 5         5 7           j = 3       6 7             j = 1     3 7               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, 137]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 137]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 6, 15, 5], X[8, 4, 9, 3], X[2, 16, 3, 15], > X[16, 7, 17, 8], X[18, 10, 19, 9], X[4, 17, 1, 18], X[12, 20, 5, 19], > X[20, 12, 13, 11], X[10, 14, 11, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -7}, {2, -1, 5, -3, 6, -10, 9, -8}, > {10, -2, 4, -5, 7, -6, 8, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-2 4 2 3 4 5 6 7 8 -7 - q + - + 11 q - 12 q + 14 q - 12 q + 10 q - 5 q + 3 q - q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 2 2 4 6 8 10 12 14 16 18 -q + -- + 3 q - 2 q + 4 q + 2 q + 4 q + 6 q + 2 q + 6 q + q + 4 q 22 24 > q - q
In[8]:=	HOMFLYPT[Link[10, Alternating, 137]][a, z]
Out[8]=	2 2 4 4 4 -4 1 2 1 2 2 z 3 z 4 z 3 z 2 z 1 - a + ----- - ----- + ----- - z - ---- + ---- - z - -- + ---- + ---- + 6 2 4 2 2 2 6 4 6 4 2 a z a z a z a a a a a 6 6 z z > -- + -- 4 2 a a
In[9]:=	Kauffman[Link[10, Alternating, 137]][a, z]
Out[9]=	3 4 -2 1 2 1 2 2 4 z 4 z 2 1 - -- - -- - a + ----- + ----- + ----- - ---- - ---- + --- + --- + 2 z + 6 4 6 2 4 2 2 2 5 3 5 3 a a a z a z a z a z a z a a 2 2 2 2 3 3 3 3 3 4 z 3 z 5 z 2 z 2 z 3 z 3 z z 2 z 3 4 > ---- + ---- - ---- - ---- - ---- + ---- + ---- + -- + ---- - a z - 7 z - 8 6 4 2 9 7 5 3 a a a a a a a a a 4 4 4 4 5 5 5 5 5 7 z 3 z 18 z z z 7 z 6 z 9 z 10 z 5 6 > ---- + ---- + ----- + -- + -- - ---- - ---- - ---- - ----- + a z + 4 z + 8 6 4 2 9 7 5 3 a a a a a a a a a 6 6 6 6 7 7 7 7 8 8 3 z 6 z 19 z 6 z 4 z 2 z 4 z 6 z 4 z 9 z > ---- - ---- - ----- - ---- + ---- + ---- + ---- + ---- + ---- + ---- + 8 6 4 2 7 5 3 a 6 4 a a a a a a a a a 8 9 9 5 z 2 z 2 z > ---- + ---- + ---- 2 5 3 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 3 1 4 3 q 3 5 5 2 7 q + 6 q + ----- + ----- + ---- + --- + --- + 7 q t + 5 q t + 7 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 > 7 q t + 5 q t + 7 q t + 5 q t + 7 q t + 2 q t + 3 q t + 13 6 15 6 17 7 > q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a137

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