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L11n271 L11n273

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DrawMorseLink

PD Presentation:

X6172 X3,11,4,10 X7,17,8,16 X15,5,16,8 X18,11,19,12 X22,17,9,18 X12,21,13,22 X20,13,21,14 X14,19,15,20 X2536 X9,1,10,4

Gauss Code:

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

Jones Polynomial:

- q-7 + q-6 - 2q-5 + 2q-4 - q-3 + q-2 + q-1 + 1 + 2q - q2 + q3

A2 (sl(3)) Invariant:

- q-22 - q-20 - 2q-18 - 3q-16 - 2q-14 - 2q-12 + q-10 + 2q-8 + 4q-6 + 6q-4 + 6q-2 + 7 + 5q2 + 3q4 + 2q6 + q8 + q10

HOMFLY-PT Polynomial:

a-2z-2 + 2a-2 + a-2z2 - z-2 - 3 - 4z2 - z4 - 2a2z-2 - 3a2 - a2z2 + 3a4z-2 + 6a4 + 4a4z2 + a4z4 - a6z-2 - 2a6 - a6z2

Kauffman Polynomial:

a-2z-2 - 4a-2 + 6a-2z2 - 5a-2z4 + a-2z6 - 2a-1z-1 + 4a-1z - 4a-1z5 + a-1z7 + z-2 - 3 + 5z2 + 3z4 - 5z6 + z8 + 2az-1 - 10az + 20az3 - 13az5 + 2az7 - 2a2z-2 + 4a2 - 3a2z2 + 7a2z4 - 6a2z6 + a2z8 + 10a3z-1 - 34a3z + 42a3z3 - 21a3z5 + 3a3z7 - 3a4z-2 + 5a4 - 4a4z2 + 5a4z4 - 5a4z6 + a4z8 + 8a5z-1 - 27a5z + 33a5z3 - 18a5z5 + 3a5z7 - a6z-2 + a6 - 2a6z2 + 6a6z4 - 5a6z6 + a6z8 + 2a7z-1 - 7a7z + 11a7z3 - 6a7z5 + a7z7

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                       1 j = 5                     1 1 j = 3                   1     j = 1               3 1 1     j = -1             2 5 1       j = -3           1   1         j = -5         1 3 2           j = -7       2 1               j = -9       1 1               j = -11   1 2                   j = -13                         j = -15 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, 272]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 272]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 11, 4, 10], X[7, 17, 8, 16], X[15, 5, 16, 8], > X[18, 11, 19, 12], X[22, 17, 9, 18], X[12, 21, 13, 22], X[20, 13, 21, 14], > X[14, 19, 15, 20], 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}, > {-11, 2, 5, -7, 8, -9, -4, 3, 6, -5, 9, -8, 7, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-7 -6 2 2 -3 -2 1 2 3 1 - q + q - -- + -- - q + q + - + 2 q - q + q 5 4 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -20 2 3 2 2 -10 2 4 6 6 2 7 - q - q - --- - --- - --- - --- + q + -- + -- + -- + -- + 5 q + 18 16 14 12 8 6 4 2 q q q q q q q q 4 6 8 10 > 3 q + 2 q + q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 272]][a, z]
Out[8]=	2 4 6 2 2 2 4 6 -2 1 2 a 3 a a 2 z -3 + -- - 3 a + 6 a - 2 a - z + ----- - ---- + ---- - -- - 4 z + -- - 2 2 2 2 2 2 2 a a z z z z a 2 2 4 2 6 2 4 4 4 > a z + 4 a z - a z - z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 272]][a, z]
Out[9]=	2 4 6 4 2 4 6 -2 1 2 a 3 a a 2 2 a -3 - -- + 4 a + 5 a + a + z + ----- - ---- - ---- - -- - --- + --- + 2 2 2 2 2 2 a z z a a z z z z 3 5 7 10 a 8 a 2 a 4 z 3 5 7 2 > ----- + ---- + ---- + --- - 10 a z - 34 a z - 27 a z - 7 a z + 5 z + z z z a 2 6 z 2 2 4 2 6 2 3 3 3 5 3 > ---- - 3 a z - 4 a z - 2 a z + 20 a z + 42 a z + 33 a z + 2 a 4 5 7 3 4 5 z 2 4 4 4 6 4 4 z 5 > 11 a z + 3 z - ---- + 7 a z + 5 a z + 6 a z - ---- - 13 a z - 2 a a 6 3 5 5 5 7 5 6 z 2 6 4 6 6 6 > 21 a z - 18 a z - 6 a z - 5 z + -- - 6 a z - 5 a z - 5 a z + 2 a 7 z 7 3 7 5 7 7 7 8 2 8 4 8 6 8 > -- + 2 a z + 3 a z + 3 a z + a z + z + a z + a z + a z a
In[10]:=	Kh[L][q, t]
Out[10]=	-3 5 1 1 2 1 2 1 1 q + - + 3 q + ------ + ------ + ------ + ----- + ----- + ----- + ----- + q 15 7 11 6 11 5 9 4 7 4 9 3 7 3 q t q t q t q t q t q t q t 1 3 1 2 2 t 2 3 2 5 3 > ----- + ----- + ----- + ---- + --- + - + q t + q t + q t + q t + 5 3 5 2 3 2 5 q t q q t q t q t q t 5 4 7 4 > q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n272

L11n271 L11n273