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L11n95 L11n97

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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X7,16,8,17 X9,18,10,19 X17,8,18,9 X22,20,5,19 X20,14,21,13 X14,22,15,21 X15,10,16,11 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

- q-15/2 + 3q-13/2 - 5q-11/2 + 6q-9/2 - 8q-7/2 + 7q-5/2 - 6q-3/2 + 4q-1/2 - 3q1/2 + q3/2

A2 (sl(3)) Invariant:

q-22 - q-20 + q-18 + q-14 + 3q-12 + 3q-8 - q-6 + q-4 + q-2 + q2 - q4

HOMFLY-PT Polynomial:

az + 3az3 + az5 - a3z-1 - 5a3z - 8a3z3 - 5a3z5 - a3z7 + a5z-1 + 2a5z + 3a5z3 + a5z5

Kauffman Polynomial:

- z2 + 3z4 - z6 + az - 9az3 + 11az5 - 3az7 - 7a2z4 + 10a2z6 - 3a2z8 - a3z-1 + 4a3z - 11a3z3 + 11a3z5 - a3z7 - a3z9 + a4 + 2a4z2 - 11a4z4 + 12a4z6 - 4a4z8 - a5z-1 + a5z + 5a5z3 - 5a5z5 + 2a5z7 - a5z9 + 3a6z2 - 4a6z4 + a6z6 - a6z8 - 2a7z + 6a7z3 - 5a7z5 + 2a8z2 - 3a8z4 - a9z3

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 4                   1 j = 2                 2   j = 0               2 1   j = -2             4 2     j = -4           4 3       j = -6         4 3         j = -8       2 4           j = -10     3 4             j = -12   1 3               j = -14   2                 j = -16 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, 96]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 96]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[7, 16, 8, 17], X[9, 18, 10, 19], > X[17, 8, 18, 9], X[22, 20, 5, 19], X[20, 14, 21, 13], X[14, 22, 15, 21], > X[15, 10, 16, 11], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -3, 5, -4, 9, 11, -2, 7, -8, -9, 3, -5, 4, > 6, -7, 8, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 3 5 6 8 7 6 4 -q + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 3 Sqrt[q] + 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -20 -18 -14 3 3 -6 -4 -2 2 4 q - q + q + q + --- + -- - q + q + q + q - q 12 8 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 96]][a, z]
Out[8]=	3 5 a a 3 5 3 3 3 5 3 5 -(--) + -- + a z - 5 a z + 2 a z + 3 a z - 8 a z + 3 a z + a z - z z 3 5 5 5 3 7 > 5 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 96]][a, z]
Out[9]=	3 5 4 a a 3 5 7 2 4 2 6 2 a - -- - -- + a z + 4 a z + a z - 2 a z - z + 2 a z + 3 a z + z z 8 2 3 3 3 5 3 7 3 9 3 4 2 4 > 2 a z - 9 a z - 11 a z + 5 a z + 6 a z - a z + 3 z - 7 a z - 4 4 6 4 8 4 5 3 5 5 5 7 5 > 11 a z - 4 a z - 3 a z + 11 a z + 11 a z - 5 a z - 5 a z - 6 2 6 4 6 6 6 7 3 7 5 7 2 8 > z + 10 a z + 12 a z + a z - 3 a z - a z + 2 a z - 3 a z - 4 8 6 8 3 9 5 9 > 4 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 4 1 2 1 3 3 4 2 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 4 2 16 6 14 5 12 5 12 4 10 4 10 3 8 3 8 2 q q q t q t q t q t q t q t q t q t 4 3 4 2 t 2 2 2 4 3 > ----- + ---- + ---- + 2 t + --- + t + 2 q t + q t 6 2 6 4 2 q t q t q t q

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