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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X4,15,1,16 X5,12,6,13 X3849 X9,16,10,17 X11,18,12,5 X17,10,18,11 X13,2,14,3

Gauss Code:

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

Jones Polynomial:

q-21/2 - q-15/2 - q-11/2 - q-7/2

A2 (sl(3)) Invariant:

- q-36 - q-34 - 2q-32 - 2q-30 - q-28 + q-26 + 3q-24 + 3q-22 + 3q-20 + 2q-18 + 2q-16 + q-14 + q-12

HOMFLY-PT Polynomial:

- 3a7z-1 - 11a7z - 15a7z3 - 7a7z5 - a7z7 + 5a9z-1 + 10a9z + 6a9z3 + a9z5 - 2a11z-1 - a11z

Kauffman Polynomial:

- 3a7z-1 + 11a7z - 15a7z3 + 7a7z5 - a7z7 + 5a8 - 10a8z2 + 6a8z4 - a8z6 - 5a9z-1 + 15a9z - 16a9z3 + 7a9z5 - a9z7 + 5a10 - 10a10z2 + 6a10z4 - a10z6 - 2a11z-1 + 4a11z - a11z3 - a14

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -6               1 j = -8               1 j = -10           1     j = -12       1         j = -14       2 1       j = -16   1             j = -18   1 1           j = -20 1               j = -22 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[9, NonAlternating, 4]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, NonAlternating, 4]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[5, 12, 6, 13], > X[3, 8, 4, 9], X[9, 16, 10, 17], X[11, 18, 12, 5], X[17, 10, 18, 11], > X[13, 2, 14, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 9, -5, -3}, {-4, -1, 2, 5, -6, 8, -7, 4, -9, -2, 3, 6, -8, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) -(15/2) -(11/2) -(7/2) q - q - q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-36 -34 2 2 -28 -26 3 3 3 2 2 -14 -q - q - --- - --- - q + q + --- + --- + --- + --- + --- + q + 32 30 24 22 20 18 16 q q q q q q q -12 > q
In[8]:=	HOMFLYPT[Link[9, NonAlternating, 4]][a, z]
Out[8]=	7 9 11 -3 a 5 a 2 a 7 9 11 7 3 9 3 ----- + ---- - ----- - 11 a z + 10 a z - a z - 15 a z + 6 a z - z z z 7 5 9 5 7 7 > 7 a z + a z - a z
In[9]:=	Kauffman[Link[9, NonAlternating, 4]][a, z]
Out[9]=	7 9 11 8 10 14 3 a 5 a 2 a 7 9 11 5 a + 5 a - a - ---- - ---- - ----- + 11 a z + 15 a z + 4 a z - z z z 8 2 10 2 7 3 9 3 11 3 8 4 10 4 > 10 a z - 10 a z - 15 a z - 16 a z - a z + 6 a z + 6 a z + 7 5 9 5 8 6 10 6 7 7 9 7 > 7 a z + 7 a z - a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-8 -6 1 1 1 1 1 2 1 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 22 7 20 7 18 6 16 6 18 5 14 4 12 4 q t q t q t q t q t q t q t 1 1 > ------ + ------ 14 3 10 2 q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L9n4

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