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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X7,16,8,17 X20,18,5,17 X18,11,19,12 X10,19,11,20 X9,14,10,15 X15,8,16,9 X2536 X4,14,1,13

Gauss Code:

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

Jones Polynomial:

2q-11/2 - 4q-9/2 + 6q-7/2 - 9q-5/2 + 8q-3/2 - 8q-1/2 + 6q1/2 - 4q3/2 + q5/2

A2 (sl(3)) Invariant:

- 2q-18 - q-16 + q-14 - 2q-12 + 2q-10 + 2q-8 + 2q-6 + 4q-4 + 3 - q2 + 2q6 - q8

HOMFLY-PT Polynomial:

a-1z3 - 2az-1 - 3az - 2az3 - az5 + 3a3z-1 + 5a3z + 3a3z3 - a5z-1 - 2a5z

Kauffman Polynomial:

- a-2z4 + 4a-1z3 - 4a-1z5 - z2 + 8z4 - 6z6 + 2az-1 - 5az + 6az3 + az5 - 4az7 - 3a2 + 11a2z4 - 7a2z6 - a2z8 + 3a3z-1 - 9a3z + 6a3z3 + 4a3z5 - 5a3z7 - 3a4 + 6a4z2 - a4z4 - a4z6 - a4z8 + a5z-1 - 4a5z + 4a5z3 - a5z5 - a5z7 - a6 + 5a6z2 - 3a6z4

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 6                 1 j = 4               3   j = 2             3 1   j = 0           5 3     j = -2         5 5       j = -4       4 3         j = -6     2 5           j = -8   2 4             j = -10   2               j = -12 2

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, NonAlternating, 33]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 33]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[7, 16, 8, 17], X[20, 18, 5, 17], > X[18, 11, 19, 12], X[10, 19, 11, 20], X[9, 14, 10, 15], X[15, 8, 16, 9], > X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, -3, 8, -7, -6, 5, -2, 10, 7, -8, 3, 4, -5, > 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	2 4 6 9 8 8 3/2 5/2 ----- - ---- + ---- - ---- + ---- - ------- + 6 Sqrt[q] - 4 q + q 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 -16 -14 2 2 2 2 4 2 6 8 3 - --- - q + q - --- + --- + -- + -- + -- - q + 2 q - q 18 12 10 8 6 4 q q q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 33]][a, z]
Out[8]=	3 5 3 -2 a 3 a a 3 5 z 3 3 3 5 ---- + ---- - -- - 3 a z + 5 a z - 2 a z + -- - 2 a z + 3 a z - a z z z z a
In[9]:=	Kauffman[Link[10, NonAlternating, 33]][a, z]
Out[9]=	3 5 2 4 6 2 a 3 a a 3 5 2 4 2 -3 a - 3 a - a + --- + ---- + -- - 5 a z - 9 a z - 4 a z - z + 6 a z + z z z 3 4 6 2 4 z 3 3 3 5 3 4 z 2 4 > 5 a z + ---- + 6 a z + 6 a z + 4 a z + 8 z - -- + 11 a z - a 2 a 5 4 4 6 4 4 z 5 3 5 5 5 6 2 6 4 6 > a z - 3 a z - ---- + a z + 4 a z - a z - 6 z - 7 a z - a z - a 7 3 7 5 7 2 8 4 8 > 4 a z - 5 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 2 2 2 4 2 5 4 3 5 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 2 12 5 10 4 8 4 8 3 6 3 6 2 4 2 4 q q t q t q t q t q t q t q t q t 5 2 2 2 4 2 6 3 > ---- + 3 t + 3 q t + q t + 3 q t + q t 2 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n33

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