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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-4 - q-3 + 4q-2 - 3q-1 + 4 - 3q + 3q2 - q3

A2 (sl(3)) Invariant:

q-14 + 2q-12 + 2q-10 + 5q-8 + 5q-6 + 4q-4 + 4q-2 + 1 + 2q2 + q6 + q8 - q10

HOMFLY-PT Polynomial:

- a-2z2 + z-2 + 2 + 2z2 + z4 - 2a2z-2 - 3a2 - 2a2z2 + a4z-2 + a4

Kauffman Polynomial:

a-3z3 - 3a-2z2 + 3a-2z4 - 3a-1z3 + 3a-1z5 - z-2 + 3 - 5z2 + 3z4 + z6 + 2az-1 - 3az - 4az3 + 4az5 - 2a2z-2 + 5a2 - 5a2z2 + a2z4 + a2z6 + 2a3z-1 - 3a3z + a3z5 - a4z-2 + 3a4 - 3a4z2 + a4z4

Khovanov Homology:

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

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