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L11n422 L11n424

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DrawMorseLink

PD Presentation:

X8192 X16,8,17,7 X10,4,11,3 X2,18,3,17 X18,9,19,10 X20,12,21,11 X5,14,6,15 X15,13,16,22 X13,6,14,1 X4,19,5,20 X12,22,7,21

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

- q-12 + q-10 + q-8 + 3q-6 + 4q-4 + 3q-2 + 5 + 3q2 + 4q4 + 3q6 + q8 + q10 - q12

HOMFLY-PT Polynomial:

a-2z-2 + a-2 - 2a-2z2 - a-2z4 - 2z-2 - 2 + 3z2 + 4z4 + z6 + a2z-2 + a2 - 2a2z2 - a2z4

Kauffman Polynomial:

3a-3z3 - 4a-3z5 + a-3z7 + a-2z-2 - 2a-2 - 7a-2z2 + 20a-2z4 - 15a-2z6 + 3a-2z8 - 2a-1z-1 + 2a-1z + a-1z3 + 5a-1z5 - 8a-1z7 + 2a-1z9 + 2z-2 - 3 - 14z2 + 40z4 - 30z6 + 6z8 - 2az-1 + 2az + az3 + 5az5 - 8az7 + 2az9 + a2z-2 - 2a2 - 7a2z2 + 20a2z4 - 15a2z6 + 3a2z8 + 3a3z3 - 4a3z5 + a3z7

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 9                 1 j = 7               2   j = 5             2 2   j = 3           3 2 1   j = 1         3 4       j = -1       2 2 1       j = -3     2 4           j = -5   1 1             j = -7   2               j = -9 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, 423]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 423]]
Out[4]=	PD[X[8, 1, 9, 2], X[16, 8, 17, 7], X[10, 4, 11, 3], X[2, 18, 3, 17], > X[18, 9, 19, 10], X[20, 12, 21, 11], X[5, 14, 6, 15], X[15, 13, 16, 22], > X[13, 6, 14, 1], X[4, 19, 5, 20], X[12, 22, 7, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -10, -7, 9}, {2, -1, 5, -3, 6, -11}, > {-9, 7, -8, -2, 4, -5, 10, -6, 11, 8}]
In[6]:=	Jones[L][q]
Out[6]=	-4 3 3 5 2 3 4 -4 - q + -- - -- + - + 5 q - 3 q + 3 q - q 3 2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -10 -8 3 4 3 2 4 6 8 10 12 5 - q + q + q + -- + -- + -- + 3 q + 4 q + 3 q + q + q - q 6 4 2 q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 423]][a, z]
Out[8]=	2 2 4 -2 2 2 1 a 2 2 z 2 2 4 z 2 4 6 -2 + a + a - -- + ----- + -- + 3 z - ---- - 2 a z + 4 z - -- - a z + z 2 2 2 2 2 2 z a z z a a
In[9]:=	Kauffman[Link[11, NonAlternating, 423]][a, z]
Out[9]=	2 2 2 2 2 1 a 2 2 a 2 z 2 7 z -3 - -- - 2 a + -- + ----- + -- - --- - --- + --- + 2 a z - 14 z - ---- - 2 2 2 2 2 a z z a 2 a z a z z a 3 3 4 5 2 2 3 z z 3 3 3 4 20 z 2 4 4 z > 7 a z + ---- + -- + a z + 3 a z + 40 z + ----- + 20 a z - ---- + 3 a 2 3 a a a 5 6 7 7 5 z 5 3 5 6 15 z 2 6 z 8 z 7 > ---- + 5 a z - 4 a z - 30 z - ----- - 15 a z + -- - ---- - 8 a z + a 2 3 a a a 8 9 3 7 8 3 z 2 8 2 z 9 > a z + 6 z + ---- + 3 a z + ---- + 2 a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	1 3 1 2 1 1 2 4 2 2 - + 4 q + 3 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 9 5 7 4 5 4 5 3 3 3 3 2 2 q t q t q t q t q t q t q t q t 3 q 3 5 3 2 5 2 7 2 9 3 > --- + 2 q t + 2 q t + q t + 2 q t + 2 q t + q t t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n423

L11n422 L11n424