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L11n445 L11n447

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DrawMorseLink

PD Presentation:

X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X18,12,19,11 X22,19,17,20 X16,21,9,22 X20,15,21,16 X12,18,13,17 X2536 X9,1,10,4

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-9/2 - 2q-7/2 + 2q-5/2 - 2q-3/2 - q-1/2 - 2q1/2 - 2q3/2 + q5/2 - 3q7/2 + q9/2 - q11/2

A2 (sl(3)) Invariant:

- q-14 - q-10 + q-6 + 3q-4 + 8q-2 + 9 + 12q2 + 11q4 + 10q6 + 10q8 + 7q10 + 6q12 + 3q14 + 2q16 + q18

HOMFLY-PT Polynomial:

- a-5z-3 - 2a-5z-1 - a-5z + 3a-3z-3 + 8a-3z-1 + 7a-3z + 2a-3z3 - 3a-1z-3 - 11a-1z-1 - 12a-1z - 6a-1z3 - a-1z5 + az-3 + 6az-1 + 8az + 5az3 + az5 - a3z-1 - 2a3z - a3z3

Kauffman Polynomial:

a-5z-3 - 5a-5z-1 + 11a-5z - 12a-5z3 + 6a-5z5 - a-5z7 - 3a-4z-2 + 9a-4 - 9a-4z2 + 4a-4z6 - a-4z8 + 3a-3z-3 - 14a-3z-1 + 34a-3z - 42a-3z3 + 24a-3z5 - 4a-3z7 - 6a-2z-2 + 21a-2 - 24a-2z2 + 6a-2z4 + 4a-2z6 - a-2z8 + 3a-1z-3 - 18a-1z-1 + 44a-1z - 57a-1z3 + 29a-1z5 - 4a-1z7 - 3z-2 + 18 - 26z2 + 8z4 + 4z6 - z8 + az-3 - 11az-1 + 27az - 36az3 + 20az5 - 3az7 + 6a2 - 14a2z2 + 6a2z4 + 3a2z6 - a2z8 - 2a3z-1 + 6a3z - 9a3z3 + 9a3z5 - 2a3z7 + a4 - 3a4z2 + 4a4z4 - a4z6

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 12                       1 j = 10                         j = 8                   3 1   j = 6               1 1 2     j = 4               2 1       j = 2           5 2 1         j = 0         2 7 2           j = -2       1 1 3             j = -4     1 2 1               j = -6   1 1                   j = -8   1                     j = -10 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, 446]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 446]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 11, 4, 10], X[7, 15, 8, 14], X[13, 5, 14, 8], > X[18, 12, 19, 11], X[22, 19, 17, 20], X[16, 21, 9, 22], X[20, 15, 21, 16], > X[12, 18, 13, 17], X[2, 5, 3, 6], X[9, 1, 10, 4]]
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]=	-(9/2) 2 2 2 1 3/2 5/2 7/2 q - ---- + ---- - ---- - ------- - 2 Sqrt[q] - 2 q + q - 3 q + 7/2 5/2 3/2 Sqrt[q] q q q 9/2 11/2 > q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -10 -6 3 8 2 4 6 8 10 9 - q - q + q + -- + -- + 12 q + 11 q + 10 q + 10 q + 7 q + 4 2 q q 12 14 16 18 > 6 q + 3 q + 2 q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 446]][a, z]
Out[8]=	3 1 3 3 a 2 8 11 6 a a z 7 z 12 z -(-----) + ----- - ---- + -- - ---- + ---- - --- + --- - -- - -- + --- - ---- + 5 3 3 3 3 3 5 3 a z z z 5 3 a a z a z a z z a z a z a a 3 3 5 3 2 z 6 z 3 3 3 z 5 > 8 a z - 2 a z + ---- - ---- + 5 a z - a z - -- + a z 3 a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 446]][a, z]
Out[9]=	9 21 2 4 1 3 3 a 3 3 6 18 + -- + -- + 6 a + a + ----- + ----- + ---- + -- - -- - ----- - ----- - 4 2 5 3 3 3 3 3 2 4 2 2 2 a a a z a z a z z z a z a z 3 5 14 18 11 a 2 a 11 z 34 z 44 z 3 > ---- - ---- - --- - ---- - ---- + ---- + ---- + ---- + 27 a z + 6 a z - 5 3 a z z z 5 3 a a z a z a a 2 2 3 3 3 2 9 z 24 z 2 2 4 2 12 z 42 z 57 z > 26 z - ---- - ----- - 14 a z - 3 a z - ----- - ----- - ----- - 4 2 5 3 a a a a a 4 5 5 3 3 3 4 6 z 2 4 4 4 6 z 24 z > 36 a z - 9 a z + 8 z + ---- + 6 a z + 4 a z + ---- + ----- + 2 5 3 a a a 5 6 6 7 29 z 5 3 5 6 4 z 4 z 2 6 4 6 z > ----- + 20 a z + 9 a z + 4 z + ---- + ---- + 3 a z - a z - -- - a 4 2 5 a a a 7 7 8 8 4 z 4 z 7 3 7 8 z z 2 8 > ---- - ---- - 3 a z - 2 a z - z - -- - -- - a z 3 a 4 2 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 2 1 1 1 1 1 2 1 2 7 + -- + 5 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + 2 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t q q t q t q t q t q t q t q t 1 1 2 2 2 4 2 6 2 4 3 6 3 > ---- + ---- + 2 t + 2 q t + q t + 2 q t + q t + q t + q t + 4 2 q t q t 6 4 8 4 8 5 12 6 > 2 q t + 3 q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n446

L11n445 L11n447