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Knotscape

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DrawMorseLink

PD Presentation:

X8192 X10,3,11,4 X14,6,15,5 X16,11,17,12 X18,13,7,14 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, -4, 6, -5}}

Jones Polynomial:

q-17/2 - 2q-15/2 + 4q-13/2 - 6q-11/2 + 6q-9/2 - 7q-7/2 + 5q-5/2 - 4q-3/2 + 2q-1/2 - q1/2

A2 (sl(3)) Invariant:

- q-26 - q-22 - 2q-20 + q-18 + 3q-14 + 3q-12 + 2q-10 + 3q-8 - q-6 + q-4 + q2

HOMFLY-PT Polynomial:

- 2az - az3 - 2a3z-1 - a3z + 2a3z3 + a3z5 + 3a5z-1 + 4a5z + 3a5z3 + a5z5 - a7z-1 - 2a7z - a7z3

Kauffman Polynomial:

- 2az + 3az3 - az5 - 2a2z2 + 5a2z4 - 2a2z6 - 2a3z-1 + 5a3z - 3a3z3 + 4a3z5 - 2a3z7 + 3a4 - 6a4z2 + 6a4z4 - a4z6 - a4z8 - 3a5z-1 + 14a5z - 22a5z3 + 15a5z5 - 5a5z7 + 3a6 - 11a6z2 + 8a6z4 - 2a6z6 - a6z8 - a7z-1 + 7a7z - 13a7z3 + 8a7z5 - 3a7z7 + a8 - 5a8z2 + 6a8z4 - 3a8z6 + 3a9z3 - 2a9z5 + 2a10z2 - a10z4

Khovanov Homology:

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

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