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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-4 - 3q-3 + 7q-2 - 10q-1 + 14 - 13q + 14q2 - 10q3 + 7q4 - 4q5 + q6

A2 (sl(3)) Invariant:

q-12 - q-10 + 2q-8 + 3q-6 + 6q-2 + 2 + 5q2 + 5q4 + 2q6 + 5q8 - 2q10 - 2q16 + q18

HOMFLY-PT Polynomial:

- a-4 + a-4z2 + a-4z4 + a-2z-2 + 4a-2 + a-2z2 - 2a-2z4 - a-2z6 - 2z-2 - 5 - 5z2 - 3z4 - z6 + a2z-2 + 2a2 + 2a2z2 + a2z4

Kauffman Polynomial:

a-6z2 - 2a-6z4 + a-6z6 + a-5z + 7a-5z3 - 11a-5z5 + 4a-5z7 - 2a-4 + a-4z2 + 6a-4z4 - 12a-4z6 + 5a-4z8 + 3a-3z + 8a-3z3 - 20a-3z5 + 5a-3z7 + 2a-3z9 + a-2z-2 - 8a-2 + 11a-2z2 - 18a-2z6 + 10a-2z8 - 2a-1z-1 + 5a-1z + 6a-1z3 - 20a-1z5 + 8a-1z7 + 2a-1z9 + 2z-2 - 9 + 20z2 - 17z4 + z6 + 5z8 - 2az-1 + 3az + 3az3 - 8az5 + 7az7 + a2z-2 - 4a2 + 8a2z2 - 8a2z4 + 6a2z6 - 2a3z3 + 3a3z5 - a4z2 + a4z4

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 = 13                     1 j = 11                   3   j = 9                 4 1   j = 7               6 3     j = 5             8 4       j = 3           6 7         j = 1         8 7           j = -1       4 8             j = -3     3 6               j = -5   1 5                 j = -7   2                   j = -9 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, 135]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 135]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[14, 5, 15, 6], X[16, 12, 17, 11], > X[10, 13, 5, 14], X[20, 18, 11, 17], X[8, 20, 9, 19], X[18, 8, 19, 7], > X[2, 9, 3, 10], X[4, 16, 1, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {3, -1, 8, -7, 9, -5}, > {4, -2, 5, -3, 10, -4, 6, -8, 7, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-4 3 7 10 2 3 4 5 6 14 + q - -- + -- - -- - 13 q + 14 q - 10 q + 7 q - 4 q + q 3 2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -10 2 3 6 2 4 6 8 10 16 18 2 + q - q + -- + -- + -- + 5 q + 5 q + 2 q + 5 q - 2 q - 2 q + q 8 6 2 q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 135]][a, z]
Out[8]=	2 2 2 4 -4 4 2 2 1 a 2 z z 2 2 4 z -5 - a + -- + 2 a - -- + ----- + -- - 5 z + -- + -- + 2 a z - 3 z + -- - 2 2 2 2 2 4 2 4 a z a z z a a a 4 6 2 z 2 4 6 z > ---- + a z - z - -- 2 2 a a
In[9]:=	Kauffman[Link[10, Alternating, 135]][a, z]
Out[9]=	2 2 8 2 2 1 a 2 2 a z 3 z 5 z -9 - -- - -- - 4 a + -- + ----- + -- - --- - --- + -- + --- + --- + 3 a z + 4 2 2 2 2 2 a z z 5 3 a a a z a z z a a 2 2 2 3 3 3 2 z z 11 z 2 2 4 2 7 z 8 z 6 z 3 > 20 z + -- + -- + ----- + 8 a z - a z + ---- + ---- + ---- + 3 a z - 6 4 2 5 3 a a a a a a 4 4 5 5 5 3 3 4 2 z 6 z 2 4 4 4 11 z 20 z 20 z > 2 a z - 17 z - ---- + ---- - 8 a z + a z - ----- - ----- - ----- - 6 4 5 3 a a a a a 6 6 6 7 7 7 5 3 5 6 z 12 z 18 z 2 6 4 z 5 z 8 z > 8 a z + 3 a z + z + -- - ----- - ----- + 6 a z + ---- + ---- + ---- + 6 4 2 5 3 a a a a a a 8 8 9 9 7 8 5 z 10 z 2 z 2 z > 7 a z + 5 z + ---- + ----- + ---- + ---- 4 2 3 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	8 1 2 1 5 3 6 4 3 - + 8 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 7 q t + 6 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 9 5 > 7 q t + 8 q t + 4 q t + 6 q t + 3 q t + 4 q t + q t + 11 5 13 6 > 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a135

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