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L10n73 L10n75

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-11 + 2q-9 - 2q-8 + 2q-7 - q-6 + 2q-5 - q-4 + q-3

A2 (sl(3)) Invariant:

q-38 + q-36 + 2q-34 + 4q-32 + 4q-30 + 4q-28 + 2q-26 + 2q-24 + q-22 + q-20 + 2q-18 + q-16 + q-14 + q-10

HOMFLY-PT Polynomial:

2a6 + 6a6z2 + 5a6z4 + a6z6 + a8z-2 + 2a8 + 6a8z2 + 5a8z4 + a8z6 - 2a10z-2 - 5a10 - 5a10z2 - a10z4 + a12z-2 + a12

Kauffman Polynomial:

- 2a6 + 6a6z2 - 5a6z4 + a6z6 + a7z + 2a7z3 - 4a7z5 + a7z7 - a8z-2 + 3a8 - 5a8z2 + 7a8z4 - 5a8z6 + a8z8 + 2a9z-1 - 8a9z + 14a9z3 - 10a9z5 + 2a9z7 - 2a10z-2 + 9a10 - 15a10z2 + 13a10z4 - 6a10z6 + a10z8 + 2a11z-1 - 8a11z + 12a11z3 - 6a11z5 + a11z7 - a12z-2 + 3a12 - 3a12z2 + a12z4 + a13z - 2a14 + a14z2

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -5                 1 j = -7               1 1 j = -9             1     j = -11         1 1 1     j = -13       1 3 1       j = -15       1 1         j = -17     2 2           j = -19 1   1             j = -21 2 1               j = -23 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[10, NonAlternating, 74]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 74]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 14, 6, 15], X[3, 8, 4, 9], X[15, 2, 16, 3], > X[16, 7, 17, 8], X[9, 18, 10, 19], X[4, 17, 1, 18], X[19, 12, 20, 5], > X[11, 20, 12, 13], X[13, 10, 14, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 4, -3, -7}, {-2, -1, 5, 3, -6, 10, -9, 8}, > {-10, 2, -4, -5, 7, 6, -8, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-11 2 2 2 -6 2 -4 -3 q + -- - -- + -- - q + -- - q + q 9 8 7 5 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-38 -36 2 4 4 4 2 2 -22 -20 2 -16 q + q + --- + --- + --- + --- + --- + --- + q + q + --- + q + 34 32 30 28 26 24 18 q q q q q q q -14 -10 > q + q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 74]][a, z]
Out[8]=	8 10 12 6 8 10 12 a 2 a a 6 2 8 2 10 2 2 a + 2 a - 5 a + a + -- - ----- + --- + 6 a z + 6 a z - 5 a z + 2 2 2 z z z 6 4 8 4 10 4 6 6 8 6 > 5 a z + 5 a z - a z + a z + a z
In[9]:=	Kauffman[Link[10, NonAlternating, 74]][a, z]
Out[9]=	8 10 12 9 11 6 8 10 12 14 a 2 a a 2 a 2 a 7 -2 a + 3 a + 9 a + 3 a - 2 a - -- - ----- - --- + ---- + ----- + a z - 2 2 2 z z z z z 9 11 13 6 2 8 2 10 2 12 2 > 8 a z - 8 a z + a z + 6 a z - 5 a z - 15 a z - 3 a z + 14 2 7 3 9 3 11 3 6 4 8 4 10 4 > a z + 2 a z + 14 a z + 12 a z - 5 a z + 7 a z + 13 a z + 12 4 7 5 9 5 11 5 6 6 8 6 10 6 > a z - 4 a z - 10 a z - 6 a z + a z - 5 a z - 6 a z + 7 7 9 7 11 7 8 8 10 8 > a z + 2 a z + a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-7 -5 1 2 1 1 1 2 2 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 23 8 21 8 19 8 21 7 19 6 17 6 17 5 q t q t q t q t q t q t q t 1 1 1 3 1 1 1 1 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 15 5 13 5 15 4 13 4 11 4 13 3 11 3 11 2 q t q t q t q t q t q t q t q t 1 1 > ----- + ---- 9 2 7 q t q t

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