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L11n275 L11n277

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X5,12,6,13 X3849 X13,2,14,3 X14,7,15,8 X9,18,10,19 X22,17,11,18 X20,11,21,12 X16,21,17,22 X4,15,1,16 X19,10,20,5

Gauss Code:

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

Jones Polynomial:

- q-12 + q-11 - q-10 + 2q-9 + q-7 + q-6 + q-4

A2 (sl(3)) Invariant:

- q-42 - q-40 - 3q-38 - 2q-36 + 2q-32 + 5q-30 + 5q-28 + 6q-26 + 5q-24 + 4q-22 + 3q-20 + 2q-18 + q-16 + q-14

HOMFLY-PT Polynomial:

2a8z-2 + 10a8 + 22a8z2 + 21a8z4 + 8a8z6 + a8z8 - 5a10z-2 - 16a10 - 19a10z2 - 8a10z4 - a10z6 + 4a12z-2 + 6a12 + 2a12z2 - a14z-2

Kauffman Polynomial:

- 2a8z-2 + 10a8 - 22a8z2 + 21a8z4 - 8a8z6 + a8z8 + 5a9z-1 - 16a9z + 19a9z3 - 8a9z5 + a9z7 - 5a10z-2 + 20a10 - 34a10z2 + 27a10z4 - 9a10z6 + a10z8 + 9a11z-1 - 27a11z + 26a11z3 - 9a11z5 + a11z7 - 4a12z-2 + 13a12 - 14a12z2 + 7a12z4 - a12z6 + 5a13z-1 - 13a13z + 8a13z3 - a13z5 - a14z-2 + 2a14 - 2a14z2 + a14z4 + a15z-1 - 2a15z + a15z3

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -7                   1 j = -9                   1 j = -11               1     j = -13           2         j = -15           2 1       j = -17       3 1           j = -19       2 1           j = -21   1 1               j = -23                     j = -25 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, 276]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 276]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 12, 6, 13], X[3, 8, 4, 9], X[13, 2, 14, 3], > X[14, 7, 15, 8], X[9, 18, 10, 19], X[22, 17, 11, 18], X[20, 11, 21, 12], > X[16, 21, 17, 22], X[4, 15, 1, 16], X[19, 10, 20, 5]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 4, -3, -10}, {-2, -1, 5, 3, -6, 11}, > {8, 2, -4, -5, 10, -9, 7, 6, -11, -8, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-12 -11 -10 2 -7 -6 -4 -q + q - q + -- + q + q + q 9 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 -40 3 2 2 5 5 6 5 4 3 2 -q - q - --- - --- + --- + --- + --- + --- + --- + --- + --- + --- + 38 36 32 30 28 26 24 22 20 18 q q q q q q q q q q -16 -14 > q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 276]][a, z]
Out[8]=	8 10 12 14 8 10 12 2 a 5 a 4 a a 8 2 10 2 10 a - 16 a + 6 a + ---- - ----- + ----- - --- + 22 a z - 19 a z + 2 2 2 2 z z z z 12 2 8 4 10 4 8 6 10 6 8 8 > 2 a z + 21 a z - 8 a z + 8 a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 276]][a, z]
Out[9]=	8 10 12 14 9 11 8 10 12 14 2 a 5 a 4 a a 5 a 9 a 10 a + 20 a + 13 a + 2 a - ---- - ----- - ----- - --- + ---- + ----- + 2 2 2 2 z z z z z z 13 15 5 a a 9 11 13 15 8 2 > ----- + --- - 16 a z - 27 a z - 13 a z - 2 a z - 22 a z - z z 10 2 12 2 14 2 9 3 11 3 13 3 > 34 a z - 14 a z - 2 a z + 19 a z + 26 a z + 8 a z + 15 3 8 4 10 4 12 4 14 4 9 5 11 5 > a z + 21 a z + 27 a z + 7 a z + a z - 8 a z - 9 a z - 13 5 8 6 10 6 12 6 9 7 11 7 8 8 10 8 > a z - 8 a z - 9 a z - a z + a z + a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-9 -7 1 1 1 2 3 1 1 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 25 9 21 8 21 7 19 6 17 6 19 5 17 5 q t q t q t q t q t q t q t 2 2 1 1 > ------ + ------ + ------ + ------ 15 4 13 4 15 3 11 2 q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n276

L11n275 L11n277