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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X11,17,12,16 X7,15,8,14 X15,9,16,8 X17,5,18,20 X13,18,14,19 X19,12,20,13 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

- q-11/2 + q-9/2 - 2q-7/2 + q-5/2 - q-3/2 + q5/2 - q7/2

A2 (sl(3)) Invariant:

q-18 + 2q-16 + 2q-14 + 2q-12 + q-10 + q-8 - q-6 - q-4 + q2 + q12

HOMFLY-PT Polynomial:

- a-3z - a-1z-1 - a-1z + 3az-1 + 6az + 5az3 + az5 - 4a3z-1 - 7a3z - 2a3z3 + 2a5z-1 + a5z

Kauffman Polynomial:

- 2a-3z + 4a-3z3 - a-3z5 + a-2 - 5a-2z2 + 5a-2z4 - a-2z6 - a-1z-1 + a-1z + a-1z3 + 3 - 9z2 + 7z4 - z6 - 3az-1 + 13az - 22az3 + 13az5 - 2az7 + 3a2 - 5a2z2 - 3a2z4 + 5a2z6 - a2z8 - 4a3z-1 + 18a3z - 30a3z3 + 18a3z5 - 3a3z7 + 2a4 - a4z2 - 5a4z4 + 5a4z6 - a4z8 - 2a5z-1 + 8a5z - 11a5z3 + 6a5z5 - a5z7

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                     1 j = 6                       j = 4               1 1 1   j = 2             1 1       j = 0           1 2 1       j = -2         1 2 2         j = -4       1 1             j = -6     1 1 1             j = -8   1 2                 j = -10                       j = -12 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]=	2
In[3]:=	Show[DrawMorseLink[Link[10, NonAlternating, 25]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 25]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[11, 17, 12, 16], X[7, 15, 8, 14], > X[15, 9, 16, 8], X[17, 5, 18, 20], X[13, 18, 14, 19], X[19, 12, 20, 13], > X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, -4, 5, 10, -2, -3, 8, -7, 4, -5, 3, -6, 7, > -8, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) -(9/2) 2 -(5/2) -(3/2) 5/2 7/2 -q + q - ---- + q - q + q - q 7/2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 2 2 2 -10 -8 -6 -4 2 12 q + --- + --- + --- + q + q - q - q + q + q 16 14 12 q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 25]][a, z]
Out[8]=	3 5 1 3 a 4 a 2 a z z 3 5 3 -(---) + --- - ---- + ---- - -- - - + 6 a z - 7 a z + a z + 5 a z - a z z z z 3 a a 3 3 5 > 2 a z + a z
In[9]:=	Kauffman[Link[10, NonAlternating, 25]][a, z]
Out[9]=	3 5 -2 2 4 1 3 a 4 a 2 a 2 z z 3 3 + a + 3 a + 2 a - --- - --- - ---- - ---- - --- + - + 13 a z + 18 a z + a z z z z 3 a a 2 3 3 5 2 5 z 2 2 4 2 4 z z 3 3 3 > 8 a z - 9 z - ---- - 5 a z - a z + ---- + -- - 22 a z - 30 a z - 2 3 a a a 4 5 5 3 4 5 z 2 4 4 4 z 5 3 5 > 11 a z + 7 z + ---- - 3 a z - 5 a z - -- + 13 a z + 18 a z + 2 3 a a 6 5 5 6 z 2 6 4 6 7 3 7 5 7 2 8 > 6 a z - z - -- + 5 a z + 5 a z - 2 a z - 3 a z - a z - a z - 2 a 4 8 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 1 2 1 1 1 1 1 2 + -- + q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ----- + 2 12 6 8 5 8 4 6 4 6 3 4 3 6 2 4 2 q q t q t q t q t q t q t q t q t 1 1 2 2 4 4 2 4 3 8 4 > ----- + - + ---- + t + q t + q t + q t + q t + q t 2 2 t 2 q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n25

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