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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X5,12,6,13 X3849 X9,14,10,15 X13,10,14,11 X11,18,12,5 X15,2,16,3

Gauss Code:

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

Jones Polynomial:

2q-19/2 - 3q-17/2 + 3q-15/2 - 4q-13/2 + 3q-11/2 - 3q-9/2 + q-7/2 - q-5/2

A2 (sl(3)) Invariant:

- q-34 - q-32 - 2q-30 - q-28 + q-26 + q-24 + 3q-22 + q-20 + 2q-18 + 2q-16 + q-14 + 2q-12 + q-8

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a5z-1 - 4a5z + 4a5z3 - a5z5 - a6 + a6z2 + 2a6z4 - a6z6 + 2a7z5 - a7z7 + 3a8 - 7a8z2 + 8a8z4 - 3a8z6 - 2a9z-1 + 6a9z - 5a9z3 + 2a9z5 - a9z7 + 5a10 - 11a10z2 + 6a10z4 - 2a10z6 - a11z-1 + 2a11z - a11z3 - a11z5 + 2a12 - 3a12z2

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           2     j = -10         1 1     j = -12       3 2       j = -14     1 2         j = -16   2 2           j = -18   1             j = -20 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[9, NonAlternating, 7]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, NonAlternating, 7]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[5, 12, 6, 13], > X[3, 8, 4, 9], X[9, 14, 10, 15], X[13, 10, 14, 11], X[11, 18, 12, 5], > X[15, 2, 16, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 9, -5, -3}, {-4, -1, 2, 5, -6, 7, -8, 4, -7, 6, -9, -2, 3, 8}]
In[6]:=	Jones[L][q]
Out[6]=	2 3 3 4 3 3 -(7/2) -(5/2) ----- - ----- + ----- - ----- + ----- - ---- + q - q 19/2 17/2 15/2 13/2 11/2 9/2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 -32 2 -28 -26 -24 3 -20 2 2 -14 2 -q - q - --- - q + q + q + --- + q + --- + --- + q + --- + 30 22 18 16 12 q q q q q -8 > q
In[8]:=	HOMFLYPT[Link[9, NonAlternating, 7]][a, z]
Out[8]=	5 9 11 a 2 a a 5 7 9 5 3 7 3 9 3 -(--) + ---- - --- - 4 a z - a z + 3 a z - 4 a z - 3 a z + a z - z z z 5 5 7 5 > a z - a z
In[9]:=	Kauffman[Link[9, NonAlternating, 7]][a, z]
Out[9]=	5 9 11 6 8 10 12 a 2 a a 5 9 11 -a + 3 a + 5 a + 2 a + -- - ---- - --- - 4 a z + 6 a z + 2 a z + z z z 6 2 8 2 10 2 12 2 5 3 9 3 11 3 > a z - 7 a z - 11 a z - 3 a z + 4 a z - 5 a z - a z + 6 4 8 4 10 4 5 5 7 5 9 5 11 5 6 6 > 2 a z + 8 a z + 6 a z - a z + 2 a z + 2 a z - a z - a z - 8 6 10 6 7 7 9 7 > 3 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 2 1 2 2 1 2 3 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 20 7 18 6 16 6 16 5 14 5 14 4 12 4 q t q t q t q t q t q t q t 2 1 1 2 1 > ------ + ------ + ------ + ----- + ---- 12 3 10 3 10 2 8 2 6 q t q t q t q t q t

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