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L11n47 L11n49

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DrawMorseLink

PD Presentation:

X6172 X18,7,19,8 X19,1,20,4 X5,14,6,15 X3849 X9,16,10,17 X15,10,16,11 X11,20,12,21 X13,22,14,5 X21,12,22,13 X2,18,3,17

Gauss Code:

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

Jones Polynomial:

q-21/2 - q-19/2 + 2q-17/2 - 2q-15/2 + 2q-13/2 - 3q-11/2 + 2q-9/2 - 2q-7/2 - q-3/2

A2 (sl(3)) Invariant:

- q-32 - q-30 - 2q-28 - 2q-26 - q-24 - q-22 + q-20 + 2q-18 + 3q-16 + 3q-14 + 2q-12 + 2q-10 + 2q-8 + q-6 + q-4

HOMFLY-PT Polynomial:

- a3z-1 - 3a3z - a3z3 - a5z-1 - 2a5z - a5z3 + 4a7z-1 + 8a7z + 5a7z3 + a7z5 - 2a9z-1 - 3a9z - a9z3

Kauffman Polynomial:

- a3z-1 + 3a3z - a3z3 + a4 - a4z2 + a5z-1 - 5a5z + 4a5z3 - a5z5 - 5a6 + 15a6z2 - 17a6z4 + 7a6z6 - a6z8 + 4a7z-1 - 15a7z + 22a7z3 - 18a7z5 + 7a7z7 - a7z9 - 6a8 + 20a8z2 - 24a8z4 + 12a8z6 - 2a8z8 + 2a9z-1 - 8a9z + 15a9z3 - 13a9z5 + 6a9z7 - a9z9 + a10 - 2a10z2 - 2a10z4 + 4a10z6 - a10z8 - a11z - 2a11z3 + 4a11z5 - a11z7 + 2a12 - 6a12z2 + 5a12z4 - a12z6

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -2                   1 j = -4                 1 2 j = -6               1   1 j = -8             2 2     j = -10           2 1       j = -12         1 3 1       j = -14       2 2           j = -16       1 1           j = -18   1 2               j = -20                     j = -22 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, 48]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 48]]
Out[4]=	PD[X[6, 1, 7, 2], X[18, 7, 19, 8], X[19, 1, 20, 4], X[5, 14, 6, 15], > X[3, 8, 4, 9], X[9, 16, 10, 17], X[15, 10, 16, 11], X[11, 20, 12, 21], > X[13, 22, 14, 5], X[21, 12, 22, 13], X[2, 18, 3, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, -5, 3}, {-4, -1, 2, 5, -6, 7, -8, 10, -9, 4, -7, 6, 11, -2, > -3, 8, -10, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) -(19/2) 2 2 2 3 2 2 -(3/2) q - q + ----- - ----- + ----- - ----- + ---- - ---- - q 17/2 15/2 13/2 11/2 9/2 7/2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 -30 2 2 -24 -22 -20 2 3 3 2 2 -q - q - --- - --- - q - q + q + --- + --- + --- + --- + --- + 28 26 18 16 14 12 10 q q q q q q q 2 -6 -4 > -- + q + q 8 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 48]][a, z]
Out[8]=	3 5 7 9 a a 4 a 2 a 3 5 7 9 3 3 5 3 -(--) - -- + ---- - ---- - 3 a z - 2 a z + 8 a z - 3 a z - a z - a z + z z z z 7 3 9 3 7 5 > 5 a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 48]][a, z]
Out[9]=	3 5 7 9 4 6 8 10 12 a a 4 a 2 a 3 5 a - 5 a - 6 a + a + 2 a - -- + -- + ---- + ---- + 3 a z - 5 a z - z z z z 7 9 11 4 2 6 2 8 2 10 2 > 15 a z - 8 a z - a z - a z + 15 a z + 20 a z - 2 a z - 12 2 3 3 5 3 7 3 9 3 11 3 6 4 > 6 a z - a z + 4 a z + 22 a z + 15 a z - 2 a z - 17 a z - 8 4 10 4 12 4 5 5 7 5 9 5 11 5 > 24 a z - 2 a z + 5 a z - a z - 18 a z - 13 a z + 4 a z + 6 6 8 6 10 6 12 6 7 7 9 7 11 7 > 7 a z + 12 a z + 4 a z - a z + 7 a z + 6 a z - a z - 6 8 8 8 10 8 7 9 9 9 > a z - 2 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 2 -2 1 1 2 1 2 1 2 q + -- + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 22 9 18 8 18 7 16 6 14 6 16 5 14 5 q q t q t q t q t q t q t q t 1 3 2 1 1 2 2 1 1 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- 12 5 12 4 10 4 12 3 10 3 8 3 8 2 6 2 4 q t q t q t q t q t q t q t q t q t

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