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Knotscape

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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X18,16,7,15 X16,12,17,11 X12,18,13,17 X6718 X4,13,5,14

Gauss Code:

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

Jones Polynomial:

- q-15/2 + 2q-13/2 - 4q-11/2 + 5q-9/2 - 6q-7/2 + 5q-5/2 - 5q-3/2 + 3q-1/2 - 2q1/2 + q3/2

A2 (sl(3)) Invariant:

q-22 + 2q-18 + q-16 + q-14 + 2q-12 + 3q-8 + q-4 - 1 - q4

HOMFLY-PT Polynomial:

az-1 + 4az + 4az3 + az5 - 3a3z-1 - 12a3z - 13a3z3 - 6a3z5 - a3z7 + 2a5z-1 + 5a5z + 4a5z3 + a5z5

Kauffman Polynomial:

1 - 4z2 + 4z4 - z6 - az-1 + 4az - 9az3 + 8az5 - 2az7 + 3a2 - 12a2z2 + 10a2z4 - a2z8 - 3a3z-1 + 14a3z - 22a3z3 + 18a3z5 - 5a3z7 + 3a4 - 10a4z2 + 11a4z4 - 2a4z6 - a4z8 - 2a5z-1 + 7a5z - 9a5z3 + 7a5z5 - 3a5z7 - a6z2 + 3a6z4 - 3a6z6 - 2a7z + 3a7z3 - 3a7z5 + a8z2 - 2a8z4 + a9z - a9z3

Khovanov Homology:

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

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