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L11n393 L11n395

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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X4,15,1,16 X9,22,10,19 X3849 X21,17,22,16 X11,5,12,18 X5,21,6,20 X17,11,18,10 X19,12,20,13 X13,2,14,3

Gauss Code:

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

Jones Polynomial:

- q-6 + 4q-5 - 4q-4 + 8q-3 - 8q-2 + 8q-1 - 6 + 5q - 3q2 + q3

A2 (sl(3)) Invariant:

- q-22 + q-20 + 4q-16 + 5q-14 + 6q-12 + 7q-10 + q-8 + 3q-6 - 2q-4 + q-2 + 1 + q4 - q6 + q8

HOMFLY-PT Polynomial:

2 + 4z2 + 4z4 + z6 + a2z-2 - 3a2 - 10a2z2 - 12a2z4 - 6a2z6 - a2z8 - 2a4z-2 + 2a4 + 7a4z2 + 5a4z4 + a4z6 + a6z-2 - a6 - a6z2

Kauffman Polynomial:

a-2z2 - 3a-2z4 + a-2z6 - 2a-1z + 6a-1z3 - 10a-1z5 + 3a-1z7 + 3 - 8z2 + 15z4 - 15z6 + 4z8 - 7az + 17az3 - 10az5 - 3az7 + 2az9 + a2z-2 + 5a2 - 28a2z2 + 51a2z4 - 35a2z6 + 8a2z8 - 2a3z-1 - 5a3z + 19a3z3 - 6a3z5 - 4a3z7 + 2a3z9 + 2a4z-2 + 2a4 - 25a4z2 + 37a4z4 - 19a4z6 + 4a4z8 - 2a5z-1 - a5z + 9a5z3 - 6a5z5 + 2a5z7 + a6z-2 - a6 - 6a6z2 + 4a6z4 - a7z + a7z3

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                   1 j = 5                 2   j = 3               3 1   j = 1             3 2     j = -1         1 6 3       j = -3         5 5         j = -5       4 4           j = -7   1 2 5             j = -9   3 3               j = -11   3                 j = -13 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 394]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 394]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[9, 22, 10, 19], > X[3, 8, 4, 9], X[21, 17, 22, 16], X[11, 5, 12, 18], X[5, 21, 6, 20], > X[17, 11, 18, 10], X[19, 12, 20, 13], X[13, 2, 14, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 11, -5, -3}, {-10, 8, -6, 4}, > {-8, -1, 2, 5, -4, 9, -7, 10, -11, -2, 3, 6, -9, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-6 4 4 8 8 8 2 3 -6 - q + -- - -- + -- - -- + - + 5 q - 3 q + q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -20 4 5 6 7 -8 3 2 -2 4 6 8 1 - q + q + --- + --- + --- + --- + q + -- - -- + q + q - q + q 16 14 12 10 6 4 q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 394]][a, z]
Out[8]=	2 4 6 2 4 6 a 2 a a 2 2 2 4 2 6 2 2 - 3 a + 2 a - a + -- - ---- + -- + 4 z - 10 a z + 7 a z - a z + 2 2 2 z z z 4 2 4 4 4 6 2 6 4 6 2 8 > 4 z - 12 a z + 5 a z + z - 6 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 394]][a, z]
Out[9]=	2 4 6 3 5 2 4 6 a 2 a a 2 a 2 a 2 z 3 3 + 5 a + 2 a - a + -- + ---- + -- - ---- - ---- - --- - 7 a z - 5 a z - 2 2 2 z z a z z z 2 3 5 7 2 z 2 2 4 2 6 2 6 z 3 > a z - a z - 8 z + -- - 28 a z - 25 a z - 6 a z + ---- + 17 a z + 2 a a 4 3 3 5 3 7 3 4 3 z 2 4 4 4 6 4 > 19 a z + 9 a z + a z + 15 z - ---- + 51 a z + 37 a z + 4 a z - 2 a 5 6 10 z 5 3 5 5 5 6 z 2 6 4 6 > ----- - 10 a z - 6 a z - 6 a z - 15 z + -- - 35 a z - 19 a z + a 2 a 7 3 z 7 3 7 5 7 8 2 8 4 8 9 > ---- - 3 a z - 4 a z + 2 a z + 4 z + 8 a z + 4 a z + 2 a z + a 3 9 > 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 6 1 3 3 1 3 2 5 4 -- + - + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 3 q 13 5 11 4 9 4 7 4 9 3 7 3 7 2 5 2 q q t q t q t q t q t q t q t q t 4 5 1 3 t 2 3 2 3 3 5 3 7 4 > ---- + ---- + --- + --- + 3 q t + 2 q t + 3 q t + q t + 2 q t + q t 5 3 q t q q t q t

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