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L11n189 L11n191

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DrawMorseLink

PD Presentation:

X8192 X12,3,13,4 X22,10,7,9 X10,14,11,13 X5,16,6,17 X20,15,21,16 X18,21,19,22 X14,19,15,20 X2738 X4,11,5,12 X17,6,18,1

Gauss Code:

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

Jones Polynomial:

q-21/2 - 3q-19/2 + 6q-17/2 - 9q-15/2 + 10q-13/2 - 12q-11/2 + 10q-9/2 - 8q-7/2 + 5q-5/2 - 2q-3/2

A2 (sl(3)) Invariant:

- q-34 - 2q-32 + q-30 - q-26 + 4q-24 + q-22 + 3q-20 + 3q-18 + 2q-14 - 3q-12 + q-10 + q-8 - 2q-6 + 2q-4

HOMFLY-PT Polynomial:

- 2a3z - 2a3z3 + a5z + a5z3 + a5z5 - 2a7z-1 - 6a7z - 4a7z3 + 3a9z-1 + 4a9z - a11z-1

Kauffman Polynomial:

2a3z - 3a3z3 + 3a4z2 - 4a4z4 - a4z6 - 3a5z3 + 2a5z5 - 3a5z7 + 2a6z2 - 2a6z4 + 2a6z6 - 3a6z8 - 2a7z-1 + 7a7z - 16a7z3 + 19a7z5 - 7a7z7 - a7z9 + 3a8 - 9a8z2 + 7a8z4 + 8a8z6 - 6a8z8 - 3a9z-1 + 12a9z - 24a9z3 + 26a9z5 - 7a9z7 - a9z9 + 3a10 - 11a10z2 + 8a10z4 + 4a10z6 - 3a10z8 - a11z-1 + 3a11z - 8a11z3 + 9a11z5 - 3a11z7 + a12 - 3a12z2 + 3a12z4 - 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                   2 j = -4                 4 1 j = -6               4 1   j = -8             6 4     j = -10           6 4       j = -12         4 6         j = -14       5 6           j = -16     2 5             j = -18   1 4               j = -20   2                 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, 190]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 190]]
Out[4]=	PD[X[8, 1, 9, 2], X[12, 3, 13, 4], X[22, 10, 7, 9], X[10, 14, 11, 13], > X[5, 16, 6, 17], X[20, 15, 21, 16], X[18, 21, 19, 22], X[14, 19, 15, 20], > X[2, 7, 3, 8], X[4, 11, 5, 12], X[17, 6, 18, 1]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10, -5, 11}, > {9, -1, 3, -4, 10, -2, 4, -8, 6, 5, -11, -7, 8, -6, 7, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 3 6 9 10 12 10 8 5 2 q - ----- + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 2 -30 -26 4 -22 3 3 2 3 -10 -8 -q - --- + q - q + --- + q + --- + --- + --- - --- + q + q - 32 24 20 18 14 12 q q q q q q 2 2 > -- + -- 6 4 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 190]][a, z]
Out[8]=	7 9 11 -2 a 3 a a 3 5 7 9 3 3 5 3 ----- + ---- - --- - 2 a z + a z - 6 a z + 4 a z - 2 a z + a z - z z z 7 3 5 5 > 4 a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 190]][a, z]
Out[9]=	7 9 11 8 10 12 2 a 3 a a 3 7 9 11 3 a + 3 a + a - ---- - ---- - --- + 2 a z + 7 a z + 12 a z + 3 a z + z z z 4 2 6 2 8 2 10 2 12 2 3 3 5 3 > 3 a z + 2 a z - 9 a z - 11 a z - 3 a z - 3 a z - 3 a z - 7 3 9 3 11 3 4 4 6 4 8 4 10 4 > 16 a z - 24 a z - 8 a z - 4 a z - 2 a z + 7 a z + 8 a z + 12 4 5 5 7 5 9 5 11 5 4 6 6 6 > 3 a z + 2 a z + 19 a z + 26 a z + 9 a z - a z + 2 a z + 8 6 10 6 12 6 5 7 7 7 9 7 11 7 > 8 a z + 4 a z - a z - 3 a z - 7 a z - 7 a z - 3 a z - 6 8 8 8 10 8 7 9 9 9 > 3 a z - 6 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 2 1 2 1 4 2 5 5 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 22 9 20 8 18 8 18 7 16 7 16 6 14 6 q q t q t q t q t q t q t q t 6 4 6 6 4 6 4 4 1 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 14 5 12 5 12 4 10 4 10 3 8 3 8 2 6 2 6 q t q t q t q t q t q t q t q t q t 4 > ---- 4 q t

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