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L11n364 L11n366

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DrawMorseLink

PD Presentation:

X6172 X10,4,11,3 X12,8,13,7 X22,15,17,16 X9,18,10,19 X17,8,18,9 X20,13,21,14 X14,21,15,22 X16,19,5,20 X2536 X4,12,1,11

Gauss Code:

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

Jones Polynomial:

q-9 - 2q-8 + 5q-7 - 6q-6 + 8q-5 - 8q-4 + 8q-3 - 5q-2 + 4q-1 - 1

A2 (sl(3)) Invariant:

q-28 + q-26 + q-24 + 4q-22 + 2q-20 + 4q-18 + 4q-16 + q-14 + 3q-12 + 3q-8 + 2q-6 + 2q-2 - 1

HOMFLY-PT Polynomial:

a2 - a2z2 - a2z4 + a4z-2 + 3a4 + 6a4z2 + 4a4z4 + a4z6 - 2a6z-2 - 6a6 - 6a6z2 - 2a6z4 + a8z-2 + 2a8 + a8z2

Kauffman Polynomial:

- az + az3 - 4a2z2 + 4a2z4 - 2a3z + 9a3z3 - 5a3z5 + 2a3z7 - a4z-2 + 7a4 - 21a4z2 + 26a4z4 - 12a4z6 + 3a4z8 + 2a5z-1 - 9a5z + 14a5z3 - 8a5z5 + a5z7 + a5z9 - 2a6z-2 + 11a6 - 23a6z2 + 25a6z4 - 17a6z6 + 5a6z8 + 2a7z-1 - 7a7z + 9a7z3 - 9a7z5 + a7z7 + a7z9 - a8z-2 + 3a8 - a8z2 - a8z4 - 4a8z6 + 2a8z8 + a9z + 3a9z3 - 6a9z5 + 2a9z7 - 2a10 + 5a10z2 - 4a10z4 + a10z6

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 j = 1                   1 j = -1                 3   j = -3               4 3   j = -5             4 1     j = -7           4 4       j = -9         4 4         j = -11       2 4           j = -13     3 4             j = -15   1 4               j = -17   1                 j = -19 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, 365]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 365]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[12, 8, 13, 7], X[22, 15, 17, 16], > X[9, 18, 10, 19], X[17, 8, 18, 9], X[20, 13, 21, 14], X[14, 21, 15, 22], > X[16, 19, 5, 20], X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {-6, 5, 9, -7, 8, -4}, > {10, -1, 3, 6, -5, -2, 11, -3, 7, -8, 4, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-9 2 5 6 8 8 8 5 4 -1 + q - -- + -- - -- + -- - -- + -- - -- + - 8 7 6 5 4 3 2 q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 -26 -24 4 2 4 4 -14 3 3 2 2 -1 + q + q + q + --- + --- + --- + --- + q + --- + -- + -- + -- 22 20 18 16 12 8 6 2 q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 365]][a, z]
Out[8]=	4 6 8 2 4 6 8 a 2 a a 2 2 4 2 6 2 8 2 a + 3 a - 6 a + 2 a + -- - ---- + -- - a z + 6 a z - 6 a z + a z - 2 2 2 z z z 2 4 4 4 6 4 4 6 > a z + 4 a z - 2 a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 365]][a, z]
Out[9]=	4 6 8 5 7 4 6 8 10 a 2 a a 2 a 2 a 3 7 a + 11 a + 3 a - 2 a - -- - ---- - -- + ---- + ---- - a z - 2 a z - 2 2 2 z z z z z 5 7 9 2 2 4 2 6 2 8 2 10 2 > 9 a z - 7 a z + a z - 4 a z - 21 a z - 23 a z - a z + 5 a z + 3 3 3 5 3 7 3 9 3 2 4 4 4 > a z + 9 a z + 14 a z + 9 a z + 3 a z + 4 a z + 26 a z + 6 4 8 4 10 4 3 5 5 5 7 5 9 5 > 25 a z - a z - 4 a z - 5 a z - 8 a z - 9 a z - 6 a z - 4 6 6 6 8 6 10 6 3 7 5 7 7 7 > 12 a z - 17 a z - 4 a z + a z + 2 a z + a z + a z + 9 7 4 8 6 8 8 8 5 9 7 9 > 2 a z + 3 a z + 5 a z + 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 1 1 1 4 3 4 2 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q q t q t q t q t q t q t q t 4 4 4 4 4 4 1 4 > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + q t 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q t q t q t q t q t q t q t q t

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