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L11n223 L11n225

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DrawMorseLink

PD Presentation:

X10,1,11,2 X3,12,4,13 X16,9,17,10 X20,12,21,11 X22,15,9,16 X5,14,6,15 X7,18,8,19 X13,4,14,5 X17,6,18,7 X19,8,20,1 X2,21,3,22

Gauss Code:

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

Jones Polynomial:

- q-25/2 + 2q-23/2 - 3q-21/2 + 3q-19/2 - 3q-17/2 + 3q-15/2 - 2q-13/2 - q-7/2

A2 (sl(3)) Invariant:

q-38 - q-36 + q-34 + q-32 + q-28 - q-26 + q-24 + q-20 + 2q-18 + q-16 + q-14 + q-12

HOMFLY-PT Polynomial:

- a7z-1 - 9a7z - 14a7z3 - 7a7z5 - a7z7 + a9z-1 + 4a9z + 2a9z3 + a11z + a11z3

Kauffman Polynomial:

- a7z-1 + 9a7z - 14a7z3 + 7a7z5 - a7z7 + a8 - 5a8z2 + 2a8z4 - a9z-1 + 5a9z - 8a9z3 + 2a9z5 - a10z2 - a10z4 - 3a11z + 6a11z3 - 3a11z5 + a12z2 + 3a12z4 - 2a12z6 - a13z + 3a13z3 + a13z5 - a13z7 - 3a14z2 + 6a14z4 - 2a14z6 - 2a15z + 3a15z3 - a15z5

Khovanov Homology:

trqj r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -6                     1 j = -8                     1 j = -10               1 1     j = -12             2         j = -14           2 2 1       j = -16         2 2           j = -18       1 2 1           j = -20     2 2               j = -22   1 2                 j = -24   1                   j = -26 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, 224]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 224]]
Out[4]=	PD[X[10, 1, 11, 2], X[3, 12, 4, 13], X[16, 9, 17, 10], X[20, 12, 21, 11], > X[22, 15, 9, 16], X[5, 14, 6, 15], X[7, 18, 8, 19], X[13, 4, 14, 5], > X[17, 6, 18, 7], X[19, 8, 20, 1], X[2, 21, 3, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, -2, 8, -6, 9, -7, 10}, > {3, -1, 4, 2, -8, 6, 5, -3, -9, 7, -10, -4, 11, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(25/2) 2 3 3 3 3 2 -(7/2) -q + ----- - ----- + ----- - ----- + ----- - ----- - q 23/2 21/2 19/2 17/2 15/2 13/2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-38 -36 -34 -32 -28 -26 -24 -20 2 -16 -14 -12 q - q + q + q + q - q + q + q + --- + q + q + q 18 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 224]][a, z]
Out[8]=	7 9 a a 7 9 11 7 3 9 3 11 3 7 5 -(--) + -- - 9 a z + 4 a z + a z - 14 a z + 2 a z + a z - 7 a z - z z 7 7 > a z
In[9]:=	Kauffman[Link[11, NonAlternating, 224]][a, z]
Out[9]=	7 9 8 a a 7 9 11 13 15 8 2 10 2 a - -- - -- + 9 a z + 5 a z - 3 a z - a z - 2 a z - 5 a z - a z + z z 12 2 14 2 7 3 9 3 11 3 13 3 15 3 > a z - 3 a z - 14 a z - 8 a z + 6 a z + 3 a z + 3 a z + 8 4 10 4 12 4 14 4 7 5 9 5 11 5 > 2 a z - a z + 3 a z + 6 a z + 7 a z + 2 a z - 3 a z + 13 5 15 5 12 6 14 6 7 7 13 7 > a z - a z - 2 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-8 -6 1 1 1 2 2 2 1 q + q + ------- + ------ + ------ + ------ + ------ + ------ + ------ + 26 10 24 9 22 9 22 8 20 8 20 7 18 7 q t q t q t q t q t q t q t 2 2 1 2 2 2 2 1 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 18 6 16 6 18 5 16 5 14 5 14 4 12 4 14 3 q t q t q t q t q t q t q t q t 1 1 > ------ + ------ 10 3 10 2 q t q t

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