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L11n141 L11n143

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DrawMorseLink

PD Presentation:

X8192 X18,11,19,12 X3,10,4,11 X17,3,18,2 X12,5,13,6 X6718 X9,16,10,17 X20,16,21,15 X22,14,7,13 X14,22,15,21 X4,20,5,19

Gauss Code:

{{1, 4, -3, -11, 5, -6}, {6, -1, -7, 3, 2, -5, 9, -10, 8, 7, -4, -2, 11, -8, 10, -9}}

Jones Polynomial:

- q-13/2 + 2q-11/2 - 4q-9/2 + 6q-7/2 - 7q-5/2 + 7q-3/2 - 7q-1/2 + 4q1/2 - 3q3/2 + q5/2

A2 (sl(3)) Invariant:

q-20 + q-16 + 2q-14 - q-12 + q-10 - q-8 - q-6 + q-4 + 4 + q2 + q4 + q6 - q8

HOMFLY-PT Polynomial:

- a-1z-1 + a-1z + a-1z3 + 2az-1 - 2az3 - az5 - 2a3z-1 - 4a3z - 3a3z3 - a3z5 + a5z-1 + 2a5z + a5z3

Kauffman Polynomial:

2a-2z2 - a-2z4 - a-1z-1 - a-1z + 6a-1z3 - 3a-1z5 + 6z2 - 7z4 + 3z6 - z8 - 2az-1 + 4az - az3 - 5az5 + 3az7 - az9 - a2 + 13a2z2 - 27a2z4 + 16a2z6 - 4a2z8 - 2a3z-1 + 10a3z - 18a3z3 + 8a3z5 - a3z9 + 8a4z2 - 16a4z4 + 11a4z6 - 3a4z8 - a5z-1 + 4a5z - 8a5z3 + 9a5z5 - 3a5z7 - a6z2 + 5a6z4 - 2a6z6 - a7z + 3a7z3 - a7z5

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 6                   1 j = 4                 2   j = 2               2 1   j = 0             5 2     j = -2           3 3       j = -4         4 4         j = -6       3 4           j = -8     1 3             j = -10   1 3               j = -12   1                 j = -14 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[11, NonAlternating, 142]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 142]]
Out[4]=	PD[X[8, 1, 9, 2], X[18, 11, 19, 12], X[3, 10, 4, 11], X[17, 3, 18, 2], > X[12, 5, 13, 6], X[6, 7, 1, 8], X[9, 16, 10, 17], X[20, 16, 21, 15], > X[22, 14, 7, 13], X[14, 22, 15, 21], X[4, 20, 5, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 4, -3, -11, 5, -6}, > {6, -1, -7, 3, 2, -5, 9, -10, 8, 7, -4, -2, 11, -8, 10, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 2 4 6 7 7 7 3/2 -q + ----- - ---- + ---- - ---- + ---- - ------- + 4 Sqrt[q] - 3 q + 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 5/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -16 2 -12 -10 -8 -6 -4 2 4 6 8 4 + q + q + --- - q + q - q - q + q + q + q + q - q 14 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 142]][a, z]
Out[8]=	3 5 3 1 2 a 2 a a z 3 5 z 3 3 3 -(---) + --- - ---- + -- + - - 4 a z + 2 a z + -- - 2 a z - 3 a z + a z z z z a a 5 3 5 3 5 > a z - a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 142]][a, z]
Out[9]=	3 5 2 1 2 a 2 a a z 3 5 7 2 -a - --- - --- - ---- - -- - - + 4 a z + 10 a z + 4 a z - a z + 6 z + a z z z z a 2 3 2 z 2 2 4 2 6 2 6 z 3 3 3 5 3 > ---- + 13 a z + 8 a z - a z + ---- - a z - 18 a z - 8 a z + 2 a a 4 5 7 3 4 z 2 4 4 4 6 4 3 z 5 > 3 a z - 7 z - -- - 27 a z - 16 a z + 5 a z - ---- - 5 a z + 2 a a 3 5 5 5 7 5 6 2 6 4 6 6 6 7 > 8 a z + 9 a z - a z + 3 z + 16 a z + 11 a z - 2 a z + 3 a z - 5 7 8 2 8 4 8 9 3 9 > 3 a z - z - 4 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 1 3 1 3 3 4 5 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 4 4 3 2 2 2 4 2 6 3 > ----- + ---- + ---- + 2 t + 2 q t + q t + 2 q t + q t 4 2 4 2 q t q t q t

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