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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X9,14,10,15 X3849 X5,11,6,10 X13,5,14,20 X11,19,12,18 X19,13,20,12 X15,2,16,3

Gauss Code:

{{1, 10, -5, -3}, {-6, -1, 2, 5, -4, 6, -8, 9, -7, 4, -10, -2, 3, 8, -9, 7}}

Jones Polynomial:

- 2q-9/2 + 2q-7/2 - 4q-5/2 + 5q-3/2 - 5q-1/2 + 4q1/2 - 3q3/2 + 2q5/2 - q7/2

A2 (sl(3)) Invariant:

q-18 + q-16 + 3q-14 + 2q-12 + 2q-10 + q-8 - 2q-6 - 2q-2 + 1 + q6 + q10

HOMFLY-PT Polynomial:

- a-1z-1 - 4a-1z - 4a-1z3 - a-1z5 + 3az-1 + 9az + 12az3 + 6az5 + az7 - 4a3z-1 - 8a3z - 5a3z3 - a3z5 + 2a5z-1 + a5z

Kauffman Polynomial:

- a-3z + 3a-3z3 - a-3z5 + a-2 - 3a-2z2 + 6a-2z4 - 2a-2z6 - a-1z-1 + 3a-1z - 5a-1z3 + 6a-1z5 - 2a-1z7 + 3 - 7z2 + 4z4 + z6 - z8 - 3az-1 + 13az - 25az3 + 16az5 - 4az7 + 3a2 - 7a2z2 + 2a2z6 - a2z8 - 4a3z-1 + 16a3z - 20a3z3 + 9a3z5 - 2a3z7 + 2a4 - 3a4z2 + 2a4z4 - a4z6 - 2a5z-1 + 7a5z - 3a5z3

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                 1 j = 6               1   j = 4             2 1   j = 2           2 1     j = 0         3 2       j = -2       3 3         j = -4     1 2           j = -6   1 3             j = -8 1 1               j = -10 2

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[10, NonAlternating, 4]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 4]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[9, 14, 10, 15], > X[3, 8, 4, 9], X[5, 11, 6, 10], X[13, 5, 14, 20], X[11, 19, 12, 18], > X[19, 13, 20, 12], X[15, 2, 16, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 10, -5, -3}, {-6, -1, 2, 5, -4, 6, -8, 9, -7, 4, -10, -2, 3, 8, > -9, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-2 2 4 5 5 3/2 5/2 7/2 ---- + ---- - ---- + ---- - ------- + 4 Sqrt[q] - 3 q + 2 q - q 9/2 7/2 5/2 3/2 Sqrt[q] q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 3 2 2 -8 2 2 6 10 1 + q + q + --- + --- + --- + q - -- - -- + q + q 14 12 10 6 2 q q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 4]][a, z]
Out[8]=	3 5 3 1 3 a 4 a 2 a 4 z 3 5 4 z 3 -(---) + --- - ---- + ---- - --- + 9 a z - 8 a z + a z - ---- + 12 a z - a z z z z a a 5 3 3 z 5 3 5 7 > 5 a z - -- + 6 a z - a z + a z a
In[9]:=	Kauffman[Link[10, NonAlternating, 4]][a, z]
Out[9]=	3 5 -2 2 4 1 3 a 4 a 2 a z 3 z 3 3 + a + 3 a + 2 a - --- - --- - ---- - ---- - -- + --- + 13 a z + 16 a z + a z z z z 3 a a 2 3 3 5 2 3 z 2 2 4 2 3 z 5 z 3 > 7 a z - 7 z - ---- - 7 a z - 3 a z + ---- - ---- - 25 a z - 2 3 a a a 4 5 5 3 3 5 3 4 6 z 4 4 z 6 z 5 > 20 a z - 3 a z + 4 z + ---- + 2 a z - -- + ---- + 16 a z + 2 3 a a a 6 7 3 5 6 2 z 2 6 4 6 2 z 7 3 7 8 2 8 > 9 a z + z - ---- + 2 a z - a z - ---- - 4 a z - 2 a z - z - a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 2 1 1 1 3 1 2 3 3 + -- + ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + 2 t + 2 10 4 8 4 8 3 6 3 6 2 4 2 4 2 q q t q t q t q t q t q t q t q t 2 2 2 4 2 4 3 6 3 8 4 > 2 q t + q t + 2 q t + q t + q t + q t

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