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L11n72 L11n74

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X16,8,17,7 X17,22,18,5 X11,18,12,19 X21,12,22,13 X13,20,14,21 X19,14,20,15 X8,16,9,15 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-17/2 - 3q-15/2 + 5q-13/2 - 8q-11/2 + 9q-9/2 - 9q-7/2 + 7q-5/2 - 6q-3/2 + 3q-1/2 - q1/2

A2 (sl(3)) Invariant:

- q-26 + q-24 - q-20 + 3q-18 + 2q-14 + q-12 + 3q-8 - q-6 + 2q-4 - 1 + q2

HOMFLY-PT Polynomial:

- az - az3 - a3z-1 - 2a3z + a3z3 + a3z5 + a5z-1 + 2a5z + 2a5z3 + a5z5 - a7z - a7z3

Kauffman Polynomial:

- az + 2az3 - az5 - a2z2 + 6a2z4 - 3a2z6 - a3z-1 + 3a3z - 3a3z3 + 8a3z5 - 4a3z7 + a4 - 2a4z4 + 5a4z6 - 3a4z8 - a5z-1 + 4a5z - 8a5z3 + 7a5z5 - 2a5z7 - a5z9 + 2a6z2 - 12a6z4 + 10a6z6 - 4a6z8 - a7z + 2a7z3 - 5a7z5 + 2a7z7 - a7z9 + 3a8z2 - 5a8z4 + 2a8z6 - a8z8 - a9z + 5a9z3 - 3a9z5 + 2a10z2 - a10z4

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 2                   1 j = 0                 2   j = -2               4 1   j = -4             4 3     j = -6           5 3       j = -8         4 4         j = -10       4 5           j = -12     2 5             j = -14   1 3               j = -16   2                 j = -18 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, 73]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 73]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[16, 8, 17, 7], X[17, 22, 18, 5], > X[11, 18, 12, 19], X[21, 12, 22, 13], X[13, 20, 14, 21], X[19, 14, 20, 15], > X[8, 16, 9, 15], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, -5, 6, -7, 8, 9, -3, -4, 5, > -8, 7, -6, 4}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 3 5 8 9 9 7 6 3 q - ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- - Sqrt[q] 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 -24 -20 3 2 -12 3 -6 2 2 -1 - q + q - q + --- + --- + q + -- - q + -- + q 18 14 8 4 q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 73]][a, z]
Out[8]=	3 5 a a 3 5 7 3 3 3 5 3 7 3 -(--) + -- - a z - 2 a z + 2 a z - a z - a z + a z + 2 a z - a z + z z 3 5 5 5 > a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 73]][a, z]
Out[9]=	3 5 4 a a 3 5 7 9 2 2 6 2 a - -- - -- - a z + 3 a z + 4 a z - a z - a z - a z + 2 a z + z z 8 2 10 2 3 3 3 5 3 7 3 9 3 > 3 a z + 2 a z + 2 a z - 3 a z - 8 a z + 2 a z + 5 a z + 2 4 4 4 6 4 8 4 10 4 5 3 5 > 6 a z - 2 a z - 12 a z - 5 a z - a z - a z + 8 a z + 5 5 7 5 9 5 2 6 4 6 6 6 8 6 > 7 a z - 5 a z - 3 a z - 3 a z + 5 a z + 10 a z + 2 a z - 3 7 5 7 7 7 4 8 6 8 8 8 5 9 7 9 > 4 a z - 2 a z + 2 a z - 3 a z - 4 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 4 1 2 1 3 2 5 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 18 7 16 6 14 6 14 5 12 5 12 4 10 4 q q q t q t q t q t q t q t q t 5 4 4 5 3 4 t 2 2 > ------ + ----- + ----- + ----- + ---- + ---- + 2 t + -- + q t 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q

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