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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X15,1,16,4 X5,12,6,13 X3849 X9,16,10,17 X11,18,12,5 X17,10,18,11 X2,14,3,13

Gauss Code:

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

Jones Polynomial:

q-17/2 - q-15/2 + 2q-13/2 - 3q-11/2 + 3q-9/2 - 3q-7/2 + q-5/2 - 2q-3/2

A2 (sl(3)) Invariant:

- q-26 - q-24 - 2q-22 - 2q-20 + 2q-14 + 2q-12 + 3q-10 + 4q-8 + 2q-6 + 2q-4

HOMFLY-PT Polynomial:

- 3a3z-1 - 6a3z - 2a3z3 + 5a5z-1 + 9a5z + 5a5z3 + a5z5 - 2a7z-1 - 3a7z - a7z3

Kauffman Polynomial:

- 3a3z-1 + 8a3z - 3a3z3 + 5a4 - 8a4z2 + 4a4z4 - a4z6 - 5a5z-1 + 13a5z - 13a5z3 + 5a5z5 - a5z7 + 5a6 - 12a6z2 + 7a6z4 - 2a6z6 - 2a7z-1 + 5a7z - 8a7z3 + 4a7z5 - a7z7 - a8z2 + 2a8z4 - a8z6 + 2a9z3 - a9z5 - a10 + 3a10z2 - a10z4

Khovanov Homology:

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

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L9n5

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