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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-19/2 - q-17/2 + 2q-15/2 - 3q-13/2 + q-11/2 - 2q-9/2 + q-7/2 - q-5/2

A2 (sl(3)) Invariant:

- q-34 - q-32 - q-30 - q-28 + 2q-22 + 2q-20 + 3q-18 + 3q-16 + q-14 + q-12 + q-8

HOMFLY-PT Polynomial:

- a5z-1 - 3a5z - 4a5z3 - a5z5 - 3a7z - 4a7z3 - a7z5 + 2a9z-1 + 4a9z + a9z3 - a11z-1

Kauffman Polynomial:

a5z-1 - 3a5z + 4a5z3 - a5z5 - a6 + 3a6z4 - a6z6 + a7z - 4a7z3 + 4a7z5 - a7z7 + 3a8 - 9a8z2 + 8a8z4 - 2a8z6 - 2a9z-1 + 5a9z - 9a9z3 + 5a9z5 - a9z7 + 5a10 - 10a10z2 + 5a10z4 - a10z6 - a11z-1 + a11z - a11z3 + 2a12 - a12z2

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -4               1 j = -6             1 1 j = -8           1     j = -10       1 1 1     j = -12       3 1       j = -14       1         j = -16   1 2           j = -18                 j = -20 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, 1]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, NonAlternating, 1]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[5, 10, 6, 11], > X[3, 8, 4, 9], X[11, 18, 12, 5], X[17, 12, 18, 13], X[9, 16, 10, 17], > X[13, 2, 14, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 9, -5, -3}, {-4, -1, 2, 5, -8, 4, -6, 7, -9, -2, 3, 8, -7, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) -(17/2) 2 3 -(11/2) 2 -(7/2) -(5/2) q - q + ----- - ----- + q - ---- + q - q 15/2 13/2 9/2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 -32 -30 -28 2 2 3 3 -14 -12 -8 -q - q - q - q + --- + --- + --- + --- + q + q + q 22 20 18 16 q q q q
In[8]:=	HOMFLYPT[Link[9, NonAlternating, 1]][a, z]
Out[8]=	5 9 11 a 2 a a 5 7 9 5 3 7 3 9 3 -(--) + ---- - --- - 3 a z - 3 a z + 4 a z - 4 a z - 4 a z + a z - z z z 5 5 7 5 > a z - a z
In[9]:=	Kauffman[Link[9, NonAlternating, 1]][a, z]
Out[9]=	5 9 11 6 8 10 12 a 2 a a 5 7 9 11 -a + 3 a + 5 a + 2 a + -- - ---- - --- - 3 a z + a z + 5 a z + a z - z z z 8 2 10 2 12 2 5 3 7 3 9 3 11 3 > 9 a z - 10 a z - a z + 4 a z - 4 a z - 9 a z - a z + 6 4 8 4 10 4 5 5 7 5 9 5 6 6 > 3 a z + 8 a z + 5 a z - a z + 4 a z + 5 a z - a z - 8 6 10 6 7 7 9 7 > 2 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 1 2 1 3 1 1 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 20 7 16 6 16 5 14 4 12 4 10 4 12 3 q t q t q t q t q t q t q t 1 1 1 1 > ------ + ------ + ----- + ---- 10 3 10 2 8 2 6 q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L9n1

L9a55 L9n2