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L11n408 L11n410

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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X7,16,8,17 X22,17,19,18 X20,12,21,11 X10,20,11,19 X18,21,5,22 X9,14,10,15 X15,8,16,9 X2536 X4,14,1,13

Gauss Code:

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

Jones Polynomial:

- 2q-5 + 5q-4 - 8q-3 + 13q-2 - 13q-1 + 14 - 11q + 9q2 - 4q3 + q4

A2 (sl(3)) Invariant:

- 2q-16 + q-12 - 2q-10 + 5q-8 + 3q-6 + 4q-4 + 6q-2 + 1 + 6q2 + 3q6 + 3q8 - 2q10 + q12

HOMFLY-PT Polynomial:

a-2z-2 + 2a-2 + a-2z2 + a-2z4 - 2z-2 - 6 - 6z2 - 3z4 - z6 + a2z-2 + 6a2 + 7a2z2 + 3a2z4 - 2a4 - 2a4z2

Kauffman Polynomial:

a-4z4 - a-3z3 + 4a-3z5 + a-2z-2 - 4a-2 + 6a-2z2 - 11a-2z4 + 9a-2z6 - 2a-1z-1 + 2a-1z + 3a-1z3 - 13a-1z5 + 10a-1z7 + 2z-2 - 11 + 26z2 - 28z4 + 6z6 + 5z8 - 2az-1 + 6az + 5az3 - 25az5 + 13az7 + az9 + a2z-2 - 12a2 + 30a2z2 - 25a2z4 - a2z6 + 6a2z8 + 6a3z - 5a3z3 - 5a3z5 + 3a3z7 + a3z9 - 4a4 + 10a4z2 - 9a4z4 + 2a4z6 + a4z8 + 2a5z - 6a5z3 + 3a5z5

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 9                   1 j = 7                 3   j = 5               6 1   j = 3             5 3     j = 1           9 6       j = -1         8 9         j = -3       5 5           j = -5     3 8             j = -7   2 5               j = -9   3                 j = -11 2

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, 409]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 409]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[7, 16, 8, 17], X[22, 17, 19, 18], > X[20, 12, 21, 11], X[10, 20, 11, 19], X[18, 21, 5, 22], X[9, 14, 10, 15], > X[15, 8, 16, 9], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {6, -5, 7, -4}, > {10, -1, -3, 9, -8, -6, 5, -2, 11, 8, -9, 3, 4, -7}]
In[6]:=	Jones[L][q]
Out[6]=	2 5 8 13 13 2 3 4 14 - -- + -- - -- + -- - -- - 11 q + 9 q - 4 q + q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 -12 2 5 3 4 6 2 6 8 10 12 1 - --- + q - --- + -- + -- + -- + -- + 6 q + 3 q + 3 q - 2 q + q 16 10 8 6 4 2 q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 409]][a, z]
Out[8]=	2 2 2 2 4 2 1 a 2 z 2 2 4 2 -6 + -- + 6 a - 2 a - -- + ----- + -- - 6 z + -- + 7 a z - 2 a z - 2 2 2 2 2 2 a z a z z a 4 4 z 2 4 6 > 3 z + -- + 3 a z - z 2 a
In[9]:=	Kauffman[Link[11, NonAlternating, 409]][a, z]
Out[9]=	2 4 2 4 2 1 a 2 2 a 2 z 3 -11 - -- - 12 a - 4 a + -- + ----- + -- - --- - --- + --- + 6 a z + 6 a z + 2 2 2 2 2 a z z a a z a z z 2 3 3 5 2 6 z 2 2 4 2 z 3 z 3 > 2 a z + 26 z + ---- + 30 a z + 10 a z - -- + ---- + 5 a z - 2 3 a a a 4 4 5 3 3 5 3 4 z 11 z 2 4 4 4 4 z > 5 a z - 6 a z - 28 z + -- - ----- - 25 a z - 9 a z + ---- - 4 2 3 a a a 5 6 13 z 5 3 5 5 5 6 9 z 2 6 4 6 > ----- - 25 a z - 5 a z + 3 a z + 6 z + ---- - a z + 2 a z + a 2 a 7 10 z 7 3 7 8 2 8 4 8 9 3 9 > ----- + 13 a z + 3 a z + 5 z + 6 a z + a z + a z + a z a
In[10]:=	Kh[L][q, t]
Out[10]=	9 2 3 2 5 3 8 5 5 8 - + 9 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 9 4 > 6 q t + 5 q t + 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n409

L11n408 L11n410