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L11n291 L11n293

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DrawMorseLink

PD Presentation:

X6172 X11,18,12,19 X3849 X2,16,3,15 X16,7,17,8 X9,11,10,22 X17,1,18,4 X19,5,20,10 X5,12,6,13 X21,15,22,14 X13,21,14,20

Gauss Code:

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

Jones Polynomial:

- q-4 + 3q-3 - 4q-2 + 7q-1 - 6 + 8q - 6q2 + 5q3 - 3q4 + q5

A2 (sl(3)) Invariant:

- q-12 + q-10 + 2q-6 + 5q-4 + 4q-2 + 7 + 4q2 + 3q4 + 2q6 - q8 + q10 - q12 + q16

HOMFLY-PT Polynomial:

a-4 + a-4z2 + a-2z-2 - 2a-2 - 5a-2z2 - 2a-2z4 - 2z-2 + 1 + 5z2 + 4z4 + z6 + a2z-2 - 2a2z2 - a2z4

Kauffman Polynomial:

a-6z2 - 2a-5z + 3a-5z3 + 2a-4 - 3a-4z2 + a-4z4 + a-4z6 - 7a-3z + 17a-3z3 - 14a-3z5 + 4a-3z7 + a-2z-2 + 4a-2 - 20a-2z2 + 30a-2z4 - 21a-2z6 + 5a-2z8 - 2a-1z-1 - 7a-1z + 24a-1z3 - 16a-1z5 - 2a-1z7 + 2a-1z9 + 2z-2 + 3 - 25z2 + 48z4 - 36z6 + 8z8 - 2az-1 - 3az + 14az3 - 6az5 - 5az7 + 2az9 + a2z-2 - 9a2z2 + 19a2z4 - 14a2z6 + 3a2z8 - a3z + 4a3z3 - 4a3z5 + a3z7

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 11                   1 j = 9                 2   j = 7               3 1   j = 5             4 3     j = 3           4 3 1     j = 1         4 6         j = -1       3 3 1         j = -3     2 5             j = -5   1 2               j = -7   2                 j = -9 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, 292]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 292]]
Out[4]=	PD[X[6, 1, 7, 2], X[11, 18, 12, 19], X[3, 8, 4, 9], X[2, 16, 3, 15], > X[16, 7, 17, 8], X[9, 11, 10, 22], X[17, 1, 18, 4], X[19, 5, 20, 10], > X[5, 12, 6, 13], X[21, 15, 22, 14], X[13, 21, 14, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, -3, 7}, {-9, -1, 5, 3, -6, 8}, > {-2, 9, -11, 10, 4, -5, -7, 2, -8, 11, -10, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-4 3 4 7 2 3 4 5 -6 - q + -- - -- + - + 8 q - 6 q + 5 q - 3 q + q 3 2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -10 2 5 4 2 4 6 8 10 12 16 7 - q + q + -- + -- + -- + 4 q + 3 q + 2 q - q + q - q + q 6 4 2 q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 292]][a, z]
Out[8]=	2 2 2 4 -4 2 2 1 a 2 z 5 z 2 2 4 2 z 1 + a - -- - -- + ----- + -- + 5 z + -- - ---- - 2 a z + 4 z - ---- - 2 2 2 2 2 4 2 2 a z a z z a a a 2 4 6 > a z + z
In[9]:=	Kauffman[Link[11, NonAlternating, 292]][a, z]
Out[9]=	2 2 4 2 1 a 2 2 a 2 z 7 z 7 z 3 3 + -- + -- + -- + ----- + -- - --- - --- - --- - --- - --- - 3 a z - a z - 4 2 2 2 2 2 a z z 5 3 a a a z a z z a a 2 2 2 3 3 3 2 z 3 z 20 z 2 2 3 z 17 z 24 z 3 > 25 z + -- - ---- - ----- - 9 a z + ---- + ----- + ----- + 14 a z + 6 4 2 5 3 a a a a a a 4 4 5 5 3 3 4 z 30 z 2 4 14 z 16 z 5 > 4 a z + 48 z + -- + ----- + 19 a z - ----- - ----- - 6 a z - 4 2 3 a a a a 6 6 7 7 3 5 6 z 21 z 2 6 4 z 2 z 7 3 7 > 4 a z - 36 z + -- - ----- - 14 a z + ---- - ---- - 5 a z + a z + 4 2 3 a a a a 8 9 8 5 z 2 8 2 z 9 > 8 z + ---- + 3 a z + ---- + 2 a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	1 3 1 2 1 2 2 5 3 3 - + 6 q + 4 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 9 5 7 4 5 4 5 3 3 3 3 2 2 q t q t q t q t q t q t q t q t 4 q 3 5 3 2 5 2 7 2 7 3 9 3 11 4 > --- + 3 q t + 4 q t + q t + 3 q t + 3 q t + q t + 2 q t + q t t

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