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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X16,9,17,10 X14,8,15,7 X18,15,19,16 X20,14,11,13 X10,17,5,18 X8,20,9,19 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

- q-8 + 3q-7 - 6q-6 + 11q-5 - 12q-4 + 15q-3 - 13q-2 + 11q-1 - 7 + 4q - q2

A2 (sl(3)) Invariant:

- q-26 - q-24 + 2q-22 + q-18 + 6q-16 + 2q-14 + 6q-12 + 4q-10 + 2q-8 + 4q-6 - 2q-4 + 4q-2 - q2 + 2q4 - q6

HOMFLY-PT Polynomial:

- z2 - z4 + a2z-2 + 3a2 + 5a2z2 + 3a2z4 + a2z6 - 2a4z-2 - 5a4 - 6a4z2 - 3a4z4 + a6z-2 + 3a6 + 3a6z2 - a8

Kauffman Polynomial:

- a-1z3 + a-1z5 + 3z2 - 7z4 + 4z6 + 4az3 - 10az5 + 6az7 + a2z-2 - 4a2 + 11a2z2 - 15a2z4 + 2a2z6 + 4a2z8 - 2a3z-1 + 3a3z + 10a3z3 - 24a3z5 + 12a3z7 + a3z9 + 2a4z-2 - 9a4 + 20a4z2 - 19a4z4 - a4z6 + 7a4z8 - 2a5z-1 + 5a5z + 5a5z3 - 18a5z5 + 10a5z7 + a5z9 + a6z-2 - 8a6 + 17a6z2 - 17a6z4 + 4a6z6 + 3a6z8 + 3a7z - 2a7z3 - 4a7z5 + 4a7z7 - 2a8 + 5a8z2 - 6a8z4 + 3a8z6 + a9z - 2a9z3 + a9z5

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 5                     1 j = 3                   3   j = 1                 4 1   j = -1               7 3     j = -3             7 5       j = -5           8 6         j = -7         6 9           j = -9       5 6             j = -11     2 7               j = -13   1 4                 j = -15   2                   j = -17 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, 129]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 129]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[16, 9, 17, 10], X[14, 8, 15, 7], > X[18, 15, 19, 16], X[20, 14, 11, 13], X[10, 17, 5, 18], X[8, 20, 9, 19], > X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 4, -8, 3, -7}, > {10, -2, 6, -4, 5, -3, 7, -5, 8, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-8 3 6 11 12 15 13 11 2 -7 - q + -- - -- + -- - -- + -- - -- + -- + 4 q - q 7 6 5 4 3 2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 -24 2 -18 6 2 6 4 2 4 2 4 2 -q - q + --- + q + --- + --- + --- + --- + -- + -- - -- + -- - q + 22 16 14 12 10 8 6 4 2 q q q q q q q q q 4 6 > 2 q - q
In[8]:=	HOMFLYPT[Link[10, Alternating, 129]][a, z]
Out[8]=	2 4 6 2 4 6 8 a 2 a a 2 2 2 4 2 6 2 3 a - 5 a + 3 a - a + -- - ---- + -- - z + 5 a z - 6 a z + 3 a z - 2 2 2 z z z 4 2 4 4 4 2 6 > z + 3 a z - 3 a z + a z
In[9]:=	Kauffman[Link[10, Alternating, 129]][a, z]
Out[9]=	2 4 6 3 5 2 4 6 8 a 2 a a 2 a 2 a 3 5 -4 a - 9 a - 8 a - 2 a + -- + ---- + -- - ---- - ---- + 3 a z + 5 a z + 2 2 2 z z z z z 3 7 9 2 2 2 4 2 6 2 8 2 z > 3 a z + a z + 3 z + 11 a z + 20 a z + 17 a z + 5 a z - -- + a 3 3 3 5 3 7 3 9 3 4 2 4 > 4 a z + 10 a z + 5 a z - 2 a z - 2 a z - 7 z - 15 a z - 5 4 4 6 4 8 4 z 5 3 5 5 5 > 19 a z - 17 a z - 6 a z + -- - 10 a z - 24 a z - 18 a z - a 7 5 9 5 6 2 6 4 6 6 6 8 6 7 > 4 a z + a z + 4 z + 2 a z - a z + 4 a z + 3 a z + 6 a z + 3 7 5 7 7 7 2 8 4 8 6 8 3 9 5 9 > 12 a z + 10 a z + 4 a z + 4 a z + 7 a z + 3 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 7 1 2 1 4 2 7 5 6 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 6 9 8 6 7 3 t 2 3 2 5 3 > ----- + ----- + ----- + ---- + ---- + --- + 4 q t + q t + 3 q t + q t 7 3 7 2 5 2 5 3 q q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a129

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