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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-15/2 - 3q-13/2 + 7q-11/2 - 12q-9/2 + 17q-7/2 - 20q-5/2 + 20q-3/2 - 19q-1/2 + 13q1/2 - 9q3/2 + 4q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-22 + q-20 - 2q-18 - q-16 + 2q-14 - 4q-12 + 3q-10 - q-8 + 4q-4 - q-2 + 7 + q4 + 2q6 - 2q8 + q10

HOMFLY-PT Polynomial:

- a-1z-1 - a-1z - 2a-1z3 - a-1z5 - az + 2az3 + 3az5 + az7 + 2a3z-1 + 6a3z + 7a3z3 + 4a3z5 + a3z7 - a5z-1 - 3a5z - 3a5z3 - a5z5

Kauffman Polynomial:

a-3z3 - a-3z5 + 5a-2z4 - 4a-2z6 - a-1z-1 + 3a-1z - 8a-1z3 + 14a-1z5 - 8a-1z7 + 1 + 2z2 - 9z4 + 15z6 - 9z8 - az - 7az3 + 8az5 + 3az7 - 6az9 - 3a2 + 14a2z2 - 30a2z4 + 29a2z6 - 10a2z8 - 2a2z10 + 2a3z-1 - 8a3z + 18a3z3 - 27a3z5 + 25a3z7 - 11a3z9 - 5a4 + 21a4z2 - 31a4z4 + 24a4z6 - 6a4z8 - 2a4z10 + a5z-1 - 4a5z + 11a5z3 - 12a5z5 + 11a5z7 - 5a5z9 - 2a6 + 7a6z2 - 12a6z4 + 13a6z6 - 5a6z8 - 5a7z3 + 8a7z5 - 3a7z7 - 2a8z2 + 3a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                       1 j = 6                     3   j = 4                   6 1   j = 2                 8 4     j = 0               11 5       j = -2             10 9         j = -4           10 10           j = -6         7 10             j = -8       5 10               j = -10     2 7                 j = -12   1 5                   j = -14   2                     j = -16 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, 209]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 209]]
Out[4]=	PD[X[8, 1, 9, 2], X[18, 9, 19, 10], X[6, 7, 1, 8], X[22, 15, 7, 16], > X[10, 4, 11, 3], X[16, 6, 17, 5], X[4, 12, 5, 11], X[12, 21, 13, 22], > X[20, 13, 21, 14], X[14, 19, 15, 20], X[2, 18, 3, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -7, 6, -3}, > {3, -1, 2, -5, 7, -8, 9, -10, 4, -6, 11, -2, 10, -9, 8, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 3 7 12 17 20 20 19 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 13 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 7/2 > 9 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -20 2 -16 2 4 3 -8 4 -2 4 6 7 - q + q - --- - q + --- - --- + --- - q + -- - q + q + 2 q - 18 14 12 10 4 q q q q q 8 10 > 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 209]][a, z]
Out[8]=	3 5 3 1 2 a a z 3 5 2 z 3 3 3 -(---) + ---- - -- - - - a z + 6 a z - 3 a z - ---- + 2 a z + 7 a z - a z z z a a 5 5 3 z 5 3 5 5 5 7 3 7 > 3 a z - -- + 3 a z + 4 a z - a z + a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 209]][a, z]
Out[9]=	3 5 2 4 6 1 2 a a 3 z 3 5 2 1 - 3 a - 5 a - 2 a - --- + ---- + -- + --- - a z - 8 a z - 4 a z + 2 z + a z z z a 3 3 2 2 4 2 6 2 8 2 z 8 z 3 3 3 > 14 a z + 21 a z + 7 a z - 2 a z + -- - ---- - 7 a z + 18 a z + 3 a a 4 5 3 7 3 4 5 z 2 4 4 4 6 4 > 11 a z - 5 a z - 9 z + ---- - 30 a z - 31 a z - 12 a z + 2 a 5 5 8 4 z 14 z 5 3 5 5 5 7 5 6 > 3 a z - -- + ----- + 8 a z - 27 a z - 12 a z + 8 a z + 15 z - 3 a a 6 7 4 z 2 6 4 6 6 6 8 6 8 z 7 3 7 > ---- + 29 a z + 24 a z + 13 a z - a z - ---- + 3 a z + 25 a z + 2 a a 5 7 7 7 8 2 8 4 8 6 8 9 > 11 a z - 3 a z - 9 z - 10 a z - 6 a z - 5 a z - 6 a z - 3 9 5 9 2 10 4 10 > 11 a z - 5 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	9 1 2 1 5 2 7 5 10 11 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 7 10 10 10 10 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 5 t + 8 q t + 4 q t + 6 q t + 6 3 6 2 4 2 4 2 q t q t q t q t q t 4 3 6 3 8 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a209

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