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L11n402 L11n404

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-5 - q-4 + 4q-3 - 3q-2 + 4q-1 - 2 + 2q - q2 + q4 - q5

A2 (sl(3)) Invariant:

q-16 + 2q-14 + 3q-12 + 6q-10 + 6q-8 + 7q-6 + 4q-4 + 2q-2 + 1 - 2q2 - q6 - q12 - q16

HOMFLY-PT Polynomial:

- a-4 - a-4z2 - a-2z-2 + 4z-2 + 8 + 8z2 + 5z4 + z6 - 5a2z-2 - 10a2 - 8a2z2 - 2a2z4 + 2a4z-2 + 3a4 + a4z2

Kauffman Polynomial:

2a-5z - 4a-5z3 + a-5z5 - a-4 + 4a-4z2 - 5a-4z4 + a-4z6 + a-3z-1 - 2a-3z + a-3z3 - a-3z5 - a-2z-2 + 2a-2 + a-2z2 - 3a-2z4 + 5a-1z-1 - 23a-1z + 40a-1z3 - 25a-1z5 + 4a-1z7 - 4z-2 + 17 - 33z2 + 42z4 - 24z6 + 4z8 + 9az-1 - 37az + 52az3 - 24az5 + az9 - 5a2z-2 + 24a2 - 51a2z2 + 58a2z4 - 30a2z6 + 5a2z8 + 5a3z-1 - 18a3z + 17a3z3 - a3z5 - 4a3z7 + a3z9 - 2a4z-2 + 11a4 - 21a4z2 + 18a4z4 - 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 r = 4 r = 5 j = 11                       1 j = 9                         j = 7                 1 1 1   j = 5               2 1       j = 3             2 2 1       j = 1           3 4 1         j = -1         1 2 3           j = -3       2 3               j = -5     2 1                 j = -7   1 4                   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, 403]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 403]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[11, 17, 12, 16], X[7, 20, 8, 21], > X[19, 8, 20, 9], X[21, 15, 22, 14], X[15, 19, 16, 22], X[13, 5, 14, 18], > X[17, 13, 18, 12], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {-5, 4, -6, 7}, > {10, -1, -4, 5, 11, -2, -3, 9, -8, 6, -7, 3, -9, 8}]
In[6]:=	Jones[L][q]
Out[6]=	-5 -4 4 3 4 2 4 5 -2 + q - q + -- - -- + - + 2 q - q + q - q 3 2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 2 3 6 6 7 4 2 2 6 12 16 1 + q + --- + --- + --- + -- + -- + -- + -- - 2 q - q - q - q 14 12 10 8 6 4 2 q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 403]][a, z]
Out[8]=	2 4 2 -4 2 4 4 1 5 a 2 a 2 z 2 2 8 - a - 10 a + 3 a + -- - ----- - ---- + ---- + 8 z - -- - 8 a z + 2 2 2 2 2 4 z a z z z a 4 2 4 2 4 6 > a z + 5 z - 2 a z + z
In[9]:=	Kauffman[Link[11, NonAlternating, 403]][a, z]
Out[9]=	2 4 -4 2 2 4 4 1 5 a 2 a 1 5 9 a 17 - a + -- + 24 a + 11 a - -- - ----- - ---- - ---- + ---- + --- + --- + 2 2 2 2 2 2 3 a z z a z a z z z a z 3 2 2 5 a 2 z 2 z 23 z 3 2 4 z z 2 2 > ---- + --- - --- - ---- - 37 a z - 18 a z - 33 z + ---- + -- - 51 a z - z 5 3 a 4 2 a a a a 3 3 3 4 4 4 2 4 z z 40 z 3 3 3 4 5 z 3 z > 21 a z - ---- + -- + ----- + 52 a z + 17 a z + 42 z - ---- - ---- + 5 3 a 4 2 a a a a 5 5 5 6 2 4 4 4 z z 25 z 5 3 5 6 z > 58 a z + 18 a z + -- - -- - ----- - 24 a z - a z - 24 z + -- - 5 3 a 4 a a a 7 2 6 4 6 4 z 3 7 8 2 8 4 8 9 3 9 > 30 a z - 7 a z + ---- - 4 a z + 4 z + 5 a z + a z + a z + a z a
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 1 1 4 2 1 2 3 - + 4 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 1 2 3 q 3 5 3 2 5 2 7 2 7 3 > ---- + --- + --- + q t + 2 q t + 2 q t + q t + q t + q t + q t + 2 q t t q t 7 4 11 5 > q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n403

L11n402 L11n404