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L11n446 L11n448

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X11,18,12,19 X19,17,20,22 X21,9,22,16 X15,21,16,20 X17,12,18,13 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

- q-17/2 + 2q-15/2 - 6q-13/2 + 7q-11/2 - 12q-9/2 + 9q-7/2 - 11q-5/2 + 8q-3/2 - 6q-1/2 + 2q1/2

A2 (sl(3)) Invariant:

q-28 + 3q-26 + 3q-24 + 6q-22 + 10q-20 + 8q-18 + 13q-16 + 12q-14 + 10q-12 + 10q-10 + 3q-8 + 4q-6 - q-4 - q-2 + 2 - 2q2

HOMFLY-PT Polynomial:

az-1 + 2az + 2az3 - a3z-3 - 7a3z-1 - 11a3z - 7a3z3 - 2a3z5 + 3a5z-3 + 12a5z-1 + 13a5z + 5a5z3 - 3a7z-3 - 7a7z-1 - 4a7z + a9z-3 + a9z-1

Kauffman Polynomial:

1 - 3z2 - 2az-1 + 5az - 7az3 - az5 + 6a2 - 15a2z2 + 14a2z4 - 7a2z6 + a3z-3 - 11a3z-1 + 31a3z - 45a3z3 + 35a3z5 - 11a3z7 - 3a4z-2 + 18a4 - 33a4z2 + 18a4z4 + 8a4z6 - 6a4z8 + 3a5z-3 - 18a5z-1 + 54a5z - 83a5z3 + 65a5z5 - 15a5z7 - a5z9 - 6a6z-2 + 21a6 - 27a6z2 - a6z4 + 22a6z6 - 8a6z8 + 3a7z-3 - 14a7z-1 + 38a7z - 55a7z3 + 34a7z5 - 5a7z7 - a7z9 - 3a8z-2 + 9a8 - 6a8z2 - 5a8z4 + 7a8z6 - 2a8z8 + a9z-3 - 5a9z-1 + 10a9z - 10a9z3 + 5a9z5 - 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                   2 j = 0                 4   j = -2               6 4   j = -4             5 3 1   j = -6           4 6       j = -8         8 5         j = -10       5 10           j = -12     1 2             j = -14   1 5               j = -16   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, 447]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 447]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[14, 7, 15, 8], X[8, 13, 5, 14], > X[11, 18, 12, 19], X[19, 17, 20, 22], X[21, 9, 22, 16], X[15, 21, 16, 20], > X[17, 12, 18, 13], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -4}, {-9, 5, -6, 8, -7, 6}, > {11, -2, -5, 9, 4, -3, -8, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 2 6 7 12 9 11 8 6 -q + ----- - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q > 2 Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 3 3 6 10 8 13 12 10 10 3 4 2 + 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 - 2 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 447]][a, z]
Out[8]=	3 5 7 9 3 5 7 9 a 3 a 3 a a a 7 a 12 a 7 a a 3 -(--) + ---- - ---- + -- + - - ---- + ----- - ---- + -- + 2 a z - 11 a z + 3 3 3 3 z z z z z z z z z 5 7 3 3 3 5 3 3 5 > 13 a z - 4 a z + 2 a z - 7 a z + 5 a z - 2 a z
In[9]:=	Kauffman[Link[11, NonAlternating, 447]][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 1 + 6 a + 18 a + 21 a + 9 a + -- + ---- + ---- + -- - ---- - ---- - ---- - 3 3 3 3 2 2 2 z z z z z z z 3 5 7 9 2 a 11 a 18 a 14 a 5 a 3 5 7 > --- - ----- - ----- - ----- - ---- + 5 a z + 31 a z + 54 a z + 38 a z + z z z z z 9 2 2 2 4 2 6 2 8 2 3 > 10 a z - 3 z - 15 a z - 33 a z - 27 a z - 6 a z - 7 a z - 3 3 5 3 7 3 9 3 2 4 4 4 6 4 > 45 a z - 83 a z - 55 a z - 10 a z + 14 a z + 18 a z - a z - 8 4 5 3 5 5 5 7 5 9 5 2 6 > 5 a z - a z + 35 a z + 65 a z + 34 a z + 5 a z - 7 a z + 4 6 6 6 8 6 3 7 5 7 7 7 9 7 > 8 a z + 22 a z + 7 a z - 11 a z - 15 a z - 5 a z - a z - 4 8 6 8 8 8 5 9 7 9 > 6 a z - 8 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 4 1 1 1 5 1 2 5 4 + q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 18 8 16 7 14 7 14 6 12 6 12 5 10 5 q q t q t q t q t q t q t q t 10 8 5 4 6 5 3 6 2 > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + 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 L11n447

L11n446 L11n448