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Knotscape

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DrawMorseLink

PD Presentation:

X8192 X10,3,11,4 X12,6,13,5 X14,11,7,12 X4,14,5,13 X2738 X6,9,1,10

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

- q-18 + q-14 + 2q-10 + 2q-8 + 2q-6 + 2q-4 + 1 - q2 + q6

HOMFLY-PT Polynomial:

- a-1z - az-1 + az3 + a3z-1 + a3z + a3z3 - a5z

Kauffman Polynomial:

a-1z - a-1z3 + 2z2 - 2z4 + az-1 - 2az + 2az3 - 2az5 - a2 + 2a2z2 - a2z4 - a2z6 + a3z-1 - 6a3z + 8a3z3 - 4a3z5 + 2a4z2 - a4z6 - 3a5z + 5a5z3 - 2a5z5 + 2a6z2 - a6z4

Khovanov Homology:

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

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