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L10n77 L10n79

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X5,14,6,15 X3849 X2,16,3,15 X16,7,17,8 X9,18,10,19 X11,20,12,13 X13,12,14,5 X17,1,18,4 X19,10,20,11

Gauss Code:

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

Jones Polynomial:

q-10 - q-9 + 2q-8 - 3q-7 + 4q-6 - 3q-5 + 3q-4 - q-3 + 2q-2

A2 (sl(3)) Invariant:

q-30 + q-28 + 2q-26 + 2q-24 + q-22 + 2q-20 + q-18 + 3q-16 + 3q-14 + 3q-12 + 4q-10 + 2q-8 + 2q-6

HOMFLY-PT Polynomial:

a4z-2 + 7a4 + 8a4z2 + 2a4z4 - 2a6z-2 - 11a6 - 13a6z2 - 6a6z4 - a6z6 + a8z-2 + 4a8 + 4a8z2 + a8z4

Kauffman Polynomial:

- a4z-2 + 8a4 - 11a4z2 + 3a4z4 + 2a5z-1 - 11a5z + 11a5z3 - 5a5z5 + a5z7 - 2a6z-2 + 13a6 - 23a6z2 + 17a6z4 - 6a6z6 + a6z8 + 2a7z-1 - 11a7z + 17a7z3 - 9a7z5 + 2a7z7 - a8z-2 + 5a8 - 8a8z2 + 11a8z4 - 5a8z6 + a8z8 + 4a9z3 - 3a9z5 + a9z7 + a10z2 - 2a10z4 + a10z6 - 2a11z3 + a11z5 + a12 - 3a12z2 + a12z4

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -3                 2 j = -5               1 2 j = -7             2     j = -9           1 1     j = -11         3 2       j = -13       1 2         j = -15     1 2           j = -17     1             j = -19 1 1               j = -21 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, 78]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 78]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 14, 6, 15], X[3, 8, 4, 9], X[2, 16, 3, 15], > X[16, 7, 17, 8], X[9, 18, 10, 19], X[11, 20, 12, 13], X[13, 12, 14, 5], > X[17, 1, 18, 4], X[19, 10, 20, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, -3, 9}, {-2, -1, 5, 3, -6, 10, -7, 8}, > {-8, 2, 4, -5, -9, 6, -10, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-10 -9 2 3 4 3 3 -3 2 q - q + -- - -- + -- - -- + -- - q + -- 8 7 6 5 4 2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 -28 2 2 -22 2 -18 3 3 3 4 2 2 q + q + --- + --- + q + --- + q + --- + --- + --- + --- + -- + -- 26 24 20 16 14 12 10 8 6 q q q q q q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 78]][a, z]
Out[8]=	4 6 8 4 6 8 a 2 a a 4 2 6 2 8 2 4 4 7 a - 11 a + 4 a + -- - ---- + -- + 8 a z - 13 a z + 4 a z + 2 a z - 2 2 2 z z z 6 4 8 4 6 6 > 6 a z + a z - a z
In[9]:=	Kauffman[Link[10, NonAlternating, 78]][a, z]
Out[9]=	4 6 8 5 7 4 6 8 12 a 2 a a 2 a 2 a 5 7 8 a + 13 a + 5 a + a - -- - ---- - -- + ---- + ---- - 11 a z - 11 a z - 2 2 2 z z z z z 4 2 6 2 8 2 10 2 12 2 5 3 7 3 > 11 a z - 23 a z - 8 a z + a z - 3 a z + 11 a z + 17 a z + 9 3 11 3 4 4 6 4 8 4 10 4 12 4 > 4 a z - 2 a z + 3 a z + 17 a z + 11 a z - 2 a z + a z - 5 5 7 5 9 5 11 5 6 6 8 6 10 6 5 7 > 5 a z - 9 a z - 3 a z + a z - 6 a z - 5 a z + a z + a z + 7 7 9 7 6 8 8 8 > 2 a z + a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 1 1 1 1 2 1 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 21 8 19 8 19 7 17 6 15 6 15 5 13 5 q q q t q t q t q t q t q t q t 2 3 2 1 1 2 1 > ------ + ------ + ------ + ----- + ----- + ----- + ---- 13 4 11 4 11 3 9 3 9 2 7 2 5 q t q t q t q t q t q t q t

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