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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X12,7,13,8 X13,1,14,4 X5,10,6,11 X3849 X9,14,10,5 X2,12,3,11

Gauss Code:

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

Jones Polynomial:

q-11/2 - q-9/2 + q-7/2 - 2q-5/2 + q-3/2 - 2q-1/2

A2 (sl(3)) Invariant:

- q-18 - q-16 - q-14 - q-12 + q-10 + 2q-8 + 3q-6 + 3q-4 + 2q-2 + 2

HOMFLY-PT Polynomial:

- 2az-1 - 2az + 3a3z-1 + 3a3z + a3z3 - a5z-1 - a5z

Kauffman Polynomial:

2az-1 - 3az - 3a2 + 2a2z2 - a2z4 + 3a3z-1 - 5a3z + 3a3z3 - a3z5 - 3a4 + 5a4z2 - 2a4z4 + a5z-1 - 2a5z + 3a5z3 - a5z5 - a6 + 3a6z2 - a6z4

Khovanov Homology:

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

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