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L11n420 L11n422

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DrawMorseLink

PD Presentation:

X8192 X16,8,17,7 X3,10,4,11 X17,2,18,3 X9,19,10,18 X20,12,21,11 X14,6,15,5 X22,15,13,16 X6,14,1,13 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 + 2q-3 - 2q-2 + 4q-1 - 3 + 4q - 3q2 + 3q3 - q4 + q5

A2 (sl(3)) Invariant:

- q-12 + q-6 + 2q-4 + 2q-2 + 4 + 3q2 + 3q4 + 3q6 + 2q8 + 3q10 + 2q12 + 2q14 + q16

HOMFLY-PT Polynomial:

a-4z-2 + 3a-4 + a-4z2 - 2a-2z-2 - 8a-2 - 8a-2z2 - 2a-2z4 + z-2 + 6 + 8z2 + 5z4 + z6 - a2 - 3a2z2 - a2z4

Kauffman Polynomial:

- a-6 + a-6z2 + a-5z3 - a-4z-2 + 4a-4 - 4a-4z2 + 2a-4z4 + 2a-3z-1 - 6a-3z + 7a-3z3 - 4a-3z5 + a-3z7 - 2a-2z-2 + 12a-2 - 25a-2z2 + 23a-2z4 - 11a-2z6 + 2a-2z8 + 2a-1z-1 - 8a-1z + 9a-1z3 - 2a-1z5 - 3a-1z7 + a-1z9 - z-2 + 10 - 31z2 + 39z4 - 22z6 + 4z8 - 3az + 9az3 - 3az5 - 3az7 + az9 + 2a2 - 11a2z2 + 18a2z4 - 11a2z6 + 2a2z8 - a3z + 6a3z3 - 5a3z5 + a3z7

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 11                   1 j = 9                 1 1 j = 7               2     j = 5             2 2     j = 3           2 1       j = 1         2 3         j = -1       2 1           j = -3     1 3             j = -5   1 1               j = -7   1                 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, 421]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 421]]
Out[4]=	PD[X[8, 1, 9, 2], X[16, 8, 17, 7], X[3, 10, 4, 11], X[17, 2, 18, 3], > X[9, 19, 10, 18], X[20, 12, 21, 11], X[14, 6, 15, 5], X[22, 15, 13, 16], > X[6, 14, 1, 13], 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 2 2 4 2 3 4 5 -3 - q + -- - -- + - + 4 q - 3 q + 3 q - q + q 3 2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -6 2 2 2 4 6 8 10 12 14 4 - q + q + -- + -- + 3 q + 3 q + 3 q + 2 q + 3 q + 2 q + 2 q + 4 2 q q 16 > q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 421]][a, z]
Out[8]=	2 2 3 8 2 -2 1 2 2 z 8 z 2 2 4 6 + -- - -- - a + z + ----- - ----- + 8 z + -- - ---- - 3 a z + 5 z - 4 2 4 2 2 2 4 2 a a a z a z a a 4 2 z 2 4 6 > ---- - a z + z 2 a
In[9]:=	Kauffman[Link[11, NonAlternating, 421]][a, z]
Out[9]=	-6 4 12 2 -2 1 2 2 2 6 z 8 z 10 - a + -- + -- + 2 a - z - ----- - ----- + ---- + --- - --- - --- - 4 2 4 2 2 2 3 a z 3 a a a a z a z a z a 2 2 2 3 3 3 3 2 z 4 z 25 z 2 2 z 7 z 9 z > 3 a z - a z - 31 z + -- - ---- - ----- - 11 a z + -- + ---- + ---- + 6 4 2 5 3 a a a a a a 4 4 5 5 3 3 3 4 2 z 23 z 2 4 4 z 2 z 5 > 9 a z + 6 a z + 39 z + ---- + ----- + 18 a z - ---- - ---- - 3 a z - 4 2 3 a a a a 6 7 7 3 5 6 11 z 2 6 z 3 z 7 3 7 8 > 5 a z - 22 z - ----- - 11 a z + -- - ---- - 3 a z + a z + 4 z + 2 3 a a a 8 9 2 z 2 8 z 9 > ---- + 2 a z + -- + a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 1 1 1 3 2 1 2 q 3 q + 2 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + --- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t q t q t 3 5 5 2 7 2 9 3 9 4 11 4 > q t + 2 q t + 2 q t + 2 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n421

L11n420 L11n422