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L10n40 L10n42

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DrawMorseLink

PD Presentation:

X8192 X10,4,11,3 X20,10,7,9 X2738 X4,15,5,16 X5,13,6,12 X11,16,12,17 X17,6,18,1 X14,19,15,20 X18,13,19,14

Gauss Code:

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

Jones Polynomial:

q-15/2 - 2q-13/2 + 3q-11/2 - 4q-9/2 + 4q-7/2 - 5q-5/2 + 3q-3/2 - 3q-1/2 + q1/2

A2 (sl(3)) Invariant:

- q-24 - q-22 - q-18 + q-16 + q-14 + q-12 + 3q-10 + 2q-8 + 3q-6 + q-4 + 1 - q2

HOMFLY-PT Polynomial:

az + az3 - 2a3z-1 - 6a3z - 4a3z3 - a3z5 + 3a5z-1 + 5a5z + 2a5z3 - a7z-1 - a7z

Kauffman Polynomial:

- z2 + 2az - 3az3 - 2a2z2 + a2z4 - a2z6 - 2a3z-1 + 9a3z - 12a3z3 + 7a3z5 - 2a3z7 + 3a4 - 6a4z2 + 4a4z4 + a4z6 - a4z8 - 3a5z-1 + 10a5z - 17a5z3 + 15a5z5 - 4a5z7 + 3a6 - 9a6z2 + 7a6z4 + a6z6 - a6z8 - a7z-1 + 3a7z - 8a7z3 + 8a7z5 - 2a7z7 + a8 - 4a8z2 + 4a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 j = 2                 1 j = 0               2   j = -2             2 2   j = -4           3 1     j = -6         2 3       j = -8       2 2         j = -10     1 2           j = -12   1 2             j = -14   1               j = -16 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, 41]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 41]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[20, 10, 7, 9], X[2, 7, 3, 8], > X[4, 15, 5, 16], X[5, 13, 6, 12], X[11, 16, 12, 17], X[17, 6, 18, 1], > X[14, 19, 15, 20], X[18, 13, 19, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, -5, -6, 8}, > {4, -1, 3, -2, -7, 6, 10, -9, 5, 7, -8, -10, 9, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 2 3 4 4 5 3 3 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + Sqrt[q] 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -22 -18 -16 -14 -12 3 2 3 -4 2 1 - q - q - q + q + q + q + --- + -- + -- + q - q 10 8 6 q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 41]][a, z]
Out[8]=	3 5 7 -2 a 3 a a 3 5 7 3 3 3 5 3 ----- + ---- - -- + a z - 6 a z + 5 a z - a z + a z - 4 a z + 2 a z - z z z 3 5 > a z
In[9]:=	Kauffman[Link[10, NonAlternating, 41]][a, z]
Out[9]=	3 5 7 4 6 8 2 a 3 a a 3 5 7 2 3 a + 3 a + a - ---- - ---- - -- + 2 a z + 9 a z + 10 a z + 3 a z - z - z z z 2 2 4 2 6 2 8 2 3 3 3 5 3 > 2 a z - 6 a z - 9 a z - 4 a z - 3 a z - 12 a z - 17 a z - 7 3 2 4 4 4 6 4 8 4 3 5 5 5 > 8 a z + a z + 4 a z + 7 a z + 4 a z + 7 a z + 15 a z + 7 5 2 6 4 6 6 6 8 6 3 7 5 7 7 7 > 8 a z - a z + a z + a z - a z - 2 a z - 4 a z - 2 a z - 4 8 6 8 > a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 2 1 2 2 2 2 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 2 3 3 1 2 2 > ----- + ----- + ----- + ---- + ---- + q t 6 3 6 2 4 2 4 2 q t q t q t q t q t

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L10n40 L10n42