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L11n448 L11n450

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X13,20,14,21 X16,12,17,11 X19,12,20,13 X8,16,5,15 X14,8,15,7 X22,17,19,18 X18,21,9,22 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

q-28 + 3q-26 + 4q-24 + 8q-22 + 12q-20 + 10q-18 + 12q-16 + 9q-14 + 6q-12 + 6q-10 + 2q-8 + 4q-6 + q-4 + q-2 + 3 - q2

HOMFLY-PT Polynomial:

- az-1 - az + az3 - a3z-3 - a3z-1 - 2a3z - 2a3z3 - a3z5 + 3a5z-3 + 6a5z-1 + 6a5z + 3a5z3 - 3a7z-3 - 5a7z-1 - 3a7z + a9z-3 + a9z-1

Kauffman Polynomial:

- z2 + az-1 - 5az3 - a2 - a2z2 + 3a2z4 - 3a2z6 + a3z-3 - 3a3z-1 + 8a3z - 19a3z3 + 18a3z5 - 6a3z7 - 3a4z-2 + 11a4 - 11a4z2 + 9a4z6 - 4a4z8 + 3a5z-3 - 12a5z-1 + 22a5z - 30a5z3 + 26a5z5 - 5a5z7 - a5z9 - 6a6z-2 + 24a6 - 28a6z2 + 3a6z4 + 14a6z6 - 5a6z8 + 3a7z-3 - 14a7z-1 + 27a7z - 29a7z3 + 14a7z5 - a7z9 - 3a8z-2 + 13a8 - 17a8z2 + 6a8z4 + 2a8z6 - a8z8 + a9z-3 - 6a9z-1 + 13a9z - 13a9z3 + 6a9z5 - a9z7

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 j = 2                   1 j = 0                 4   j = -2               5 4   j = -4             3 2 1   j = -6           4 5       j = -8         5 3         j = -10       4 8           j = -12     1 1             j = -14     4               j = -16 1 1                 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]=	4
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 449]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 449]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[13, 20, 14, 21], X[16, 12, 17, 11], > X[19, 12, 20, 13], X[8, 16, 5, 15], X[14, 8, 15, 7], X[22, 17, 19, 18], > X[18, 21, 9, 22], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 7, -6}, {-5, 3, 9, -8}, > {11, -2, 4, 5, -3, -7, 6, -4, 8, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) -(15/2) 5 5 9 7 8 6 5 -q + q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q > Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 3 4 8 12 10 12 9 6 6 2 4 3 + q + --- + --- + --- + --- + --- + --- + --- + --- + --- + -- + -- + 26 24 22 20 18 16 14 12 10 8 6 q q q q q q q q q q q -4 -2 2 > q + q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 449]][a, z]
Out[8]=	3 5 7 9 3 5 7 9 a 3 a 3 a a a a 6 a 5 a a 3 5 -(--) + ---- - ---- + -- - - - -- + ---- - ---- + -- - a z - 2 a z + 6 a z - 3 3 3 3 z z z z z z z z z 7 3 3 3 5 3 3 5 > 3 a z + a z - 2 a z + 3 a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 449]][a, z]
Out[9]=	3 5 7 9 4 6 8 2 4 6 8 a 3 a 3 a a 3 a 6 a 3 a a -a + 11 a + 24 a + 13 a + -- + ---- + ---- + -- - ---- - ---- - ---- + - - 3 3 3 3 2 2 2 z z z z z z z z 3 5 7 9 3 a 12 a 14 a 6 a 3 5 7 9 2 > ---- - ----- - ----- - ---- + 8 a z + 22 a z + 27 a z + 13 a z - z - z z z z 2 2 4 2 6 2 8 2 3 3 3 5 3 > a z - 11 a z - 28 a z - 17 a z - 5 a z - 19 a z - 30 a z - 7 3 9 3 2 4 6 4 8 4 3 5 5 5 > 29 a z - 13 a z + 3 a z + 3 a z + 6 a z + 18 a z + 26 a z + 7 5 9 5 2 6 4 6 6 6 8 6 3 7 > 14 a z + 6 a z - 3 a z + 9 a z + 14 a z + 2 a z - 6 a z - 5 7 9 7 4 8 6 8 8 8 5 9 7 9 > 5 a z - a z - 4 a z - 5 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 4 1 1 1 4 1 1 4 4 + q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 18 8 16 8 16 7 14 6 12 6 12 5 10 5 q q t q t q t q t q t q t q t 8 5 3 4 5 3 2 5 2 > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + q t 10 4 8 4 8 3 6 3 6 2 4 2 4 2 q t q t q t q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n449

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