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L11n321 L11n323

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X5,14,6,15 X8493 X2,16,3,15 X16,7,17,8 X9,18,10,19 X4,17,1,18 X19,13,20,22 X13,10,14,11 X21,5,22,12 X11,21,12,20

Gauss Code:

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

Jones Polynomial:

q-6 - 3q-5 + 6q-4 - 8q-3 + 11q-2 - 10q-1 + 11 - 7q + 5q2 - 2q3

A2 (sl(3)) Invariant:

q-18 - q-16 + 2q-14 + 2q-12 + 5q-8 + 2q-6 + 5q-4 + 5q-2 + 3 + 5q2 - q4 + q6 - 2q10

HOMFLY-PT Polynomial:

- 2a-2 - 2a-2z2 + z-2 + 7 + 8z2 + 3z4 - 2a2z-2 - 7a2 - 8a2z2 - 4a2z4 - a2z6 + a4z-2 + 2a4 + 2a4z2 + a4z4

Kauffman Polynomial:

- 2a-3z + 3a-3z3 + 3a-2 - 5a-2z2 + 4a-2z4 + a-2z6 - 7a-1z + 10a-1z3 - 4a-1z5 + 3a-1z7 - z-2 + 9 - 15z2 + 10z4 - 4z6 + 3z8 + 2az-1 - 10az + 17az3 - 17az5 + 6az7 + az9 - 2a2z-2 + 7a2 - 8a2z2 + 5a2z4 - 11a2z6 + 6a2z8 + 2a3z-1 - 6a3z + 17a3z3 - 22a3z5 + 6a3z7 + a3z9 - a4z-2 + a4 + 5a4z2 - 4a4z4 - 5a4z6 + 3a4z8 - a5z + 7a5z3 - 9a5z5 + 3a5z7 - a6 + 3a6z2 - 3a6z4 + a6z6

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 7                   2 j = 5                 3   j = 3               4 2   j = 1             7 3     j = -1           5 6       j = -3         6 5         j = -5       4 7           j = -7     2 4             j = -9   1 4               j = -11   2                 j = -13 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, 322]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 322]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 14, 6, 15], X[8, 4, 9, 3], X[2, 16, 3, 15], > X[16, 7, 17, 8], X[9, 18, 10, 19], X[4, 17, 1, 18], X[19, 13, 20, 22], > X[13, 10, 14, 11], X[21, 5, 22, 12], X[11, 21, 12, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -7}, {-2, -1, 5, -3, -6, 9, -11, 10}, > {-9, 2, 4, -5, 7, 6, -8, 11, -10, 8}]
In[6]:=	Jones[L][q]
Out[6]=	-6 3 6 8 11 10 2 3 11 + q - -- + -- - -- + -- - -- - 7 q + 5 q - 2 q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 2 2 5 2 5 5 2 4 6 10 3 + q - q + --- + --- + -- + -- + -- + -- + 5 q - q + q - 2 q 14 12 8 6 4 2 q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 322]][a, z]
Out[8]=	2 4 2 2 2 4 -2 2 a a 2 2 z 2 2 4 2 7 - -- - 7 a + 2 a + z - ---- + -- + 8 z - ---- - 8 a z + 2 a z + 2 2 2 2 a z z a 4 2 4 4 4 2 6 > 3 z - 4 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 322]][a, z]
Out[9]=	2 4 3 3 2 4 6 -2 2 a a 2 a 2 a 2 z 7 z 9 + -- + 7 a + a - a - z - ---- - -- + --- + ---- - --- - --- - 10 a z - 2 2 2 z z 3 a a z z a 2 3 3 3 5 2 5 z 2 2 4 2 6 2 3 z 10 z > 6 a z - a z - 15 z - ---- - 8 a z + 5 a z + 3 a z + ---- + ----- + 2 3 a a a 4 3 3 3 5 3 4 4 z 2 4 4 4 6 4 > 17 a z + 17 a z + 7 a z + 10 z + ---- + 5 a z - 4 a z - 3 a z - 2 a 5 6 4 z 5 3 5 5 5 6 z 2 6 4 6 > ---- - 17 a z - 22 a z - 9 a z - 4 z + -- - 11 a z - 5 a z + a 2 a 7 6 6 3 z 7 3 7 5 7 8 2 8 4 8 > a z + ---- + 6 a z + 6 a z + 3 a z + 3 z + 6 a z + 3 a z + a 9 3 9 > a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	6 1 2 1 4 2 4 4 7 - + 7 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 6 5 5 3 3 2 5 2 7 3 > ----- + ---- + --- + 3 q t + 4 q t + 2 q t + 3 q t + 2 q t 3 2 3 q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n322

L11n321 L11n323