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DrawMorseLink

PD Presentation:

X8192 X2,11,3,12 X12,3,13,4 X16,5,17,6 X6718 X4,15,5,16 X20,14,21,13 X18,10,19,9 X10,20,11,19 X22,18,7,17 X14,22,15,21

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 3q-11/2 - 8q-9/2 + 13q-7/2 - 18q-5/2 + 20q-3/2 - 21q-1/2 + 17q1/2 - 13q3/2 + 8q5/2 - 3q7/2 + q9/2

A2 (sl(3)) Invariant:

q-18 - q-16 + 4q-14 - q-12 + 3q-10 + 3q-8 - 2q-6 + 6q-4 - 3q-2 + 5 - q2 - 2q4 + q6 - 4q8 + q10 - q12

HOMFLY-PT Polynomial:

2a-1z-1 + 9a-1z + 10a-1z3 + 5a-1z5 + a-1z7 - 5az-1 - 21az - 29az3 - 20az5 - 7az7 - az9 + 3a3z-1 + 9a3z + 10a3z3 + 5a3z5 + a3z7

Kauffman Polynomial:

a-4 - 3a-4z2 + 3a-4z4 - a-4z6 - 4a-3z3 + 7a-3z5 - 3a-3z7 - 4a-2z4 + 10a-2z6 - 5a-2z8 + 2a-1z-1 - 9a-1z + 10a-1z3 - 5a-1z5 + 8a-1z7 - 5a-1z9 - 5 + 20z2 - 27z4 + 24z6 - 7z8 - 2z10 + 5az-1 - 25az + 41az3 - 35az5 + 25az7 - 11az9 - 5a2 + 18a2z2 - 28a2z4 + 25a2z6 - 9a2z8 - 2a2z10 + 3a3z-1 - 12a3z + 17a3z3 - 12a3z5 + 8a3z7 - 6a3z9 - 4a4z4 + 9a4z6 - 7a4z8 + 3a5z - 8a5z3 + 10a5z5 - 6a5z7 - a6z2 + 4a6z4 - 3a6z6 - a7z + 2a7z3 - a7z5

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 = 10                       1 j = 8                     2   j = 6                   6 1   j = 4                 7 2     j = 2               10 6       j = 0             11 7         j = -2           10 11           j = -4         8 10             j = -6       5 10               j = -8     3 8                 j = -10     5                   j = -12 1 3                     j = -14 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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 229]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 229]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[16, 5, 17, 6], > X[6, 7, 1, 8], X[4, 15, 5, 16], X[20, 14, 21, 13], X[18, 10, 19, 9], > X[10, 20, 11, 19], X[22, 18, 7, 17], X[14, 22, 15, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -6, 4, -5}, > {5, -1, 8, -9, 2, -3, 7, -11, 6, -4, 10, -8, 9, -7, 11, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 3 8 13 18 20 21 -q + ----- - ---- + ---- - ---- + ---- - ------- + 17 Sqrt[q] - 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 3/2 5/2 7/2 9/2 > 13 q + 8 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 4 -12 3 3 2 6 3 2 4 6 5 + q - q + --- - q + --- + -- - -- + -- - -- - q - 2 q + q - 14 10 8 6 4 2 q q q q q q 8 10 12 > 4 q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 229]][a, z]
Out[8]=	3 3 5 2 5 a 3 a 9 z 3 10 z 3 3 3 5 z --- - --- + ---- + --- - 21 a z + 9 a z + ----- - 29 a z + 10 a z + ---- - a z z z a a a 7 5 3 5 z 7 3 7 9 > 20 a z + 5 a z + -- - 7 a z + a z - a z a
In[9]:=	Kauffman[Link[11, Alternating, 229]][a, z]
Out[9]=	3 -4 2 2 5 a 3 a 9 z 3 5 7 -5 + a - 5 a + --- + --- + ---- - --- - 25 a z - 12 a z + 3 a z - a z + a z z z a 2 3 3 2 3 z 2 2 6 2 4 z 10 z 3 3 3 > 20 z - ---- + 18 a z - a z - ---- + ----- + 41 a z + 17 a z - 4 3 a a a 4 4 5 3 7 3 4 3 z 4 z 2 4 4 4 6 4 > 8 a z + 2 a z - 27 z + ---- - ---- - 28 a z - 4 a z + 4 a z + 4 2 a a 5 5 6 6 7 z 5 z 5 3 5 5 5 7 5 6 z 10 z > ---- - ---- - 35 a z - 12 a z + 10 a z - a z + 24 z - -- + ----- + 3 a 4 2 a a a 7 7 2 6 4 6 6 6 3 z 8 z 7 3 7 5 7 > 25 a z + 9 a z - 3 a z - ---- + ---- + 25 a z + 8 a z - 6 a z - 3 a a 8 9 8 5 z 2 8 4 8 5 z 9 3 9 10 > 7 z - ---- - 9 a z - 7 a z - ---- - 11 a z - 6 a z - 2 z - 2 a a 2 10 > 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	11 1 1 3 5 3 8 5 10 11 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 6 12 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 8 10 10 2 2 2 4 2 4 3 > ----- + ---- + ---- + 7 t + 10 q t + 6 q t + 7 q t + 2 q t + 4 2 4 2 q t q t q t 6 3 6 4 8 4 10 5 > 6 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a229

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