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L10n48 L10n50

Knotscape

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DrawMorseLink

PD Presentation:

X8192 X18,9,19,10 X6718 X20,14,7,13 X12,5,13,6 X3,10,4,11 X4,15,5,16 X11,16,12,17 X14,20,15,19 X17,2,18,3

Gauss Code:

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

Jones Polynomial:

- q-17/2 + q-15/2 - 2q-13/2 + 2q-11/2 - 2q-9/2 + 2q-7/2 - 2q-5/2 + q-3/2 - q-1/2

A2 (sl(3)) Invariant:

q-30 + q-26 + q-24 + q-22 + q-20 + q-16 - q-14 + q-8 + q-6 + q-4 + q-2

HOMFLY-PT Polynomial:

- a3z-1 - 6a3z - 5a3z3 - a3z5 + 2a5z-1 + 8a5z + 11a5z3 + 6a5z5 + a5z7 - 2a7z-1 - 6a7z - 5a7z3 - a7z5 + a9z-1 + a9z

Kauffman Polynomial:

- a3z-1 + 7a3z - 11a3z3 + 6a3z5 - a3z7 + a4z2 - 6a4z4 + 5a4z6 - a4z8 - 2a5z-1 + 12a5z - 24a5z3 + 16a5z5 - 3a5z7 - a6 + a6z2 - 3a6z4 + 4a6z6 - a6z8 - 2a7z-1 + 10a7z - 15a7z3 + 10a7z5 - 2a7z7 - a8z2 + 3a8z4 - a8z6 - a9z-1 + 4a9z - 2a9z3 - a10z2 - a11z

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 0                 1 j = -2                   j = -4             2 1   j = -6           1 1     j = -8         1 1       j = -10       1 1         j = -12     1 1           j = -14     1             j = -16 1 1               j = -18 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[10, NonAlternating, 49]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 49]]
Out[4]=	PD[X[8, 1, 9, 2], X[18, 9, 19, 10], X[6, 7, 1, 8], X[20, 14, 7, 13], > X[12, 5, 13, 6], X[3, 10, 4, 11], X[4, 15, 5, 16], X[11, 16, 12, 17], > X[14, 20, 15, 19], X[17, 2, 18, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 10, -6, -7, 5, -3}, > {3, -1, 2, 6, -8, -5, 4, -9, 7, 8, -10, -2, 9, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) -(15/2) 2 2 2 2 2 -(3/2) 1 -q + q - ----- + ----- - ---- + ---- - ---- + q - ------- 13/2 11/2 9/2 7/2 5/2 Sqrt[q] q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 -26 -24 -22 -20 -16 -14 -8 -6 -4 -2 q + q + q + q + q + q - q + q + q + q + q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 49]][a, z]
Out[8]=	3 5 7 9 a 2 a 2 a a 3 5 7 9 3 3 -(--) + ---- - ---- + -- - 6 a z + 8 a z - 6 a z + a z - 5 a z + z z z z 5 3 7 3 3 5 5 5 7 5 5 7 > 11 a z - 5 a z - a z + 6 a z - a z + a z
In[9]:=	Kauffman[Link[10, NonAlternating, 49]][a, z]
Out[9]=	3 5 7 9 6 a 2 a 2 a a 3 5 7 9 11 -a - -- - ---- - ---- - -- + 7 a z + 12 a z + 10 a z + 4 a z - a z + z z z z 4 2 6 2 8 2 10 2 3 3 5 3 7 3 9 3 > a z + a z - a z - a z - 11 a z - 24 a z - 15 a z - 2 a z - 4 4 6 4 8 4 3 5 5 5 7 5 4 6 > 6 a z - 3 a z + 3 a z + 6 a z + 16 a z + 10 a z + 5 a z + 6 6 8 6 3 7 5 7 7 7 4 8 6 8 > 4 a z - a z - a z - 3 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 2 1 1 1 1 1 1 1 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 18 6 16 6 16 5 14 4 12 4 12 3 10 3 q q t q t q t q t q t q t q t 1 1 1 1 t 2 > ------ + ----- + ---- + ---- + -- + t 10 2 8 2 8 6 4 q t q t q t q t q

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