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Knotscape

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DrawMorseLink

PD Presentation:

X8192 X10,3,11,4 X14,6,15,5 X18,11,7,12 X16,13,17,14 X12,17,13,18 X4,16,5,15 X2738 X6,9,1,10

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

- q-24 + q-20 - q-18 + q-16 + q-14 + 2q-10 + q-8 + 2q-6 + q-4 + 2 - q2 + q6

HOMFLY-PT Polynomial:

- a-1z - az-1 - az + az3 + a3z-1 + 2a3z + 2a3z3 + a5z3 - a7z

Kauffman Polynomial:

a-1z - a-1z3 + z2 - 2z4 + az-1 - 3az + 3az3 - 3az5 - a2 - a2z2 + 3a2z4 - 3a2z6 + a3z-1 - 5a3z + 6a3z3 - 2a3z7 - a4z2 + 5a4z4 - a4z6 - a4z8 + 2a5z - 7a5z3 + 11a5z5 - 4a5z7 - 3a6z2 + 4a6z4 + a6z6 - a6z8 + 3a7z - 9a7z3 + 8a7z5 - 2a7z7 - 4a8z2 + 4a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 4                   1 j = 2                 1   j = 0               3 1   j = -2             3 2     j = -4           3 2       j = -6         3 4         j = -8       2 2           j = -10     1 3             j = -12   1 2               j = -14   1                 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, 25]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 25]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 3, 11, 4], X[14, 6, 15, 5], X[18, 11, 7, 12], > X[16, 13, 17, 14], X[12, 17, 13, 18], X[4, 16, 5, 15], X[2, 7, 3, 8], > X[6, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 2, -7, 3, -9}, {8, -1, 9, -2, 4, -6, 5, -3, 7, -5, 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 2 3 5 5 6 5 4 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]=	-24 -20 -18 -16 -14 2 -8 2 -4 2 6 2 - q + q - q + q + q + --- + q + -- + q - q + q 10 6 q q
In[8]:=	HOMFLYPT[Link[9, Alternating, 25]][a, z]
Out[8]=	3 a a z 3 7 3 3 3 5 3 -(-) + -- - - - a z + 2 a z - a z + a z + 2 a z + a z z z a
In[9]:=	Kauffman[Link[9, Alternating, 25]][a, z]
Out[9]=	3 2 a a z 3 5 7 2 2 2 4 2 -a + - + -- + - - 3 a z - 5 a z + 2 a z + 3 a z + z - a z - a z - z z a 3 6 2 8 2 z 3 3 3 5 3 7 3 4 > 3 a z - 4 a z - -- + 3 a z + 6 a z - 7 a z - 9 a z - 2 z + a 2 4 4 4 6 4 8 4 5 5 5 7 5 > 3 a z + 5 a z + 4 a z + 4 a z - 3 a z + 11 a z + 8 a z - 2 6 4 6 6 6 8 6 3 7 5 7 7 7 4 8 > 3 a z - a z + a z - a z - 2 a z - 4 a z - 2 a z - a z - 6 8 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 2 1 3 2 2 3 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 3 4 3 2 3 2 4 2 > ----- + ----- + ----- + ---- + ---- + t + q t + q t 6 3 6 2 4 2 4 2 q t q t q t q t q t

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