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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X14,8,15,7 X16,11,5,12 X12,15,13,16 X8,14,9,13 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

q-22 + 2q-20 + q-16 + q-14 - q-12 + q-10 + q-6 + q-4 + 2 - q2 + q6

HOMFLY-PT Polynomial:

- a-1z - az-1 - az + az3 + 2a3z-1 + 3a3z + 2a3z3 - 2a5z-1 - 3a5z + a7z-1

Kauffman Polynomial:

a-1z - a-1z3 + z2 - 2z4 + az-1 - 3az + 3az3 - 3az5 - 3a2z2 + 4a2z4 - 3a2z6 + 2a3z-1 - 9a3z + 12a3z3 - 4a3z5 - a3z7 + a4 - 6a4z2 + 11a4z4 - 5a4z6 + 2a5z-1 - 8a5z + 11a5z3 - 2a5z5 - a5z7 - 2a6z2 + 5a6z4 - 2a6z6 + a7z-1 - 3a7z + 3a7z3 - a7z5

Khovanov Homology:

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

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