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L11n356 L11n358

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X11,19,12,18 X7,16,8,17 X15,8,16,9 X17,15,18,22 X13,21,14,20 X19,13,20,12 X21,5,22,14 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-5 - q-4 + 3q-3 - 3q-2 + 4q-1 - 3 + 4q - 2q2 + 2q3 - q4

A2 (sl(3)) Invariant:

q-16 + 2q-14 + 2q-12 + 3q-10 + 2q-8 + 3q-6 + 3q-4 + 3q-2 + 4 + 2q2 + 2q4 + q6 - q12

HOMFLY-PT Polynomial:

- a-2 - 3a-2z2 - a-2z4 + z-2 + 6 + 8z2 + 5z4 + z6 - 2a2z-2 - 8a2 - 8a2z2 - 2a2z4 + a4z-2 + 3a4 + a4z2

Kauffman Polynomial:

a-5z - a-4 + 2a-4z2 + a-3z - 2a-3z3 + a-3z5 - a-2 + 3a-2z2 - 6a-2z4 + 2a-2z6 - 3a-1z + 11a-1z3 - 12a-1z5 + 3a-1z7 - z-2 + 8 - 20z2 + 24z4 - 15z6 + 3z8 + 2az-1 - 10az + 18az3 - 8az5 - 2az7 + az9 - 2a2z-2 + 14a2 - 38a2z2 + 47a2z4 - 24a2z6 + 4a2z8 + 2a3z-1 - 7a3z + 5a3z3 + 5a3z5 - 5a3z7 + a3z9 - a4z-2 + 7a4 - 17a4z2 + 17a4z4 - 7a4z6 + a4z8

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 9                   1 j = 7                 1   j = 5               2 2   j = 3             2 1 1   j = 1           2 3       j = -1         2 2 1       j = -3       2 3           j = -5     1 1             j = -7   1 3               j = -9                     j = -11 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, 357]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 357]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[11, 19, 12, 18], X[7, 16, 8, 17], > X[15, 8, 16, 9], X[17, 15, 18, 22], X[13, 21, 14, 20], X[19, 13, 20, 12], > X[21, 5, 22, 14], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {-5, 4, -6, 3, -8, 7, -9, 6}, > {10, -1, -4, 5, 11, -2, -3, 8, -7, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-5 -4 3 3 4 2 3 4 -3 + q - q + -- - -- + - + 4 q - 2 q + 2 q - q 3 2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 2 2 3 2 3 3 3 2 4 6 12 4 + q + --- + --- + --- + -- + -- + -- + -- + 2 q + 2 q + q - q 14 12 10 8 6 4 2 q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 357]][a, z]
Out[8]=	2 4 2 -2 2 4 -2 2 a a 2 3 z 2 2 4 2 6 - a - 8 a + 3 a + z - ---- + -- + 8 z - ---- - 8 a z + a z + 2 2 2 z z a 4 4 z 2 4 6 > 5 z - -- - 2 a z + z 2 a
In[9]:=	Kauffman[Link[11, NonAlternating, 357]][a, z]
Out[9]=	2 4 3 -4 -2 2 4 -2 2 a a 2 a 2 a z z 3 z 8 - a - a + 14 a + 7 a - z - ---- - -- + --- + ---- + -- + -- - --- - 2 2 z z 5 3 a z z a a 2 2 3 3 2 2 z 3 z 2 2 4 2 2 z > 10 a z - 7 a z - 20 z + ---- + ---- - 38 a z - 17 a z - ---- + 4 2 3 a a a 3 4 5 11 z 3 3 3 4 6 z 2 4 4 4 z > ----- + 18 a z + 5 a z + 24 z - ---- + 47 a z + 17 a z + -- - a 2 3 a a 5 6 7 12 z 5 3 5 6 2 z 2 6 4 6 3 z > ----- - 8 a z + 5 a z - 15 z + ---- - 24 a z - 7 a z + ---- - a 2 a a 7 3 7 8 2 8 4 8 9 3 9 > 2 a z - 5 a z + 3 z + 4 a z + a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	1 3 1 1 3 1 1 2 3 - + 3 q + 2 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 11 6 7 5 7 4 5 4 5 3 3 3 3 2 q t q t q t q t q t q t q t 2 2 2 q 3 5 3 2 5 2 7 2 9 3 > ---- + --- + --- + q t + 2 q t + q t + 2 q t + q t + q t 2 q t t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n357

L11n356 L11n358