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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-15/2 - 4q-13/2 + 8q-11/2 - 14q-9/2 + 19q-7/2 - 22q-5/2 + 22q-3/2 - 20q-1/2 + 14q1/2 - 9q3/2 + 4q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-22 + 2q-20 - q-18 + q-16 + 4q-14 - 3q-12 + 4q-10 - 2q-8 - q-6 + 2q-4 - 3q-2 + 6 - q2 + q4 + 2q6 - 2q8 + q10

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a-3z3 - a-3z5 + 5a-2z4 - 4a-2z6 - a-1z-1 + 2a-1z - 6a-1z3 + 13a-1z5 - 8a-1z7 + 5z2 - 13z4 + 18z6 - 10z8 - 2az-1 + 6az - 6az3 - 3az5 + 11az7 - 8az9 - a2 + 14a2z2 - 38a2z4 + 34a2z6 - 9a2z8 - 3a2z10 - 2a3z-1 + 6a3z + 6a3z3 - 35a3z5 + 38a3z7 - 15a3z9 + 13a4z2 - 35a4z4 + 31a4z6 - 6a4z8 - 3a4z10 - a5z-1 + 2a5z - 8a5z5 + 15a5z7 - 7a5z9 + 4a6z2 - 13a6z4 + 18a6z6 - 7a6z8 - 5a7z3 + 10a7z5 - 4a7z7 + 2a8z4 - 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 3     j = 0               12 6       j = -2             11 9         j = -4           11 11           j = -6         9 12             j = -8       5 10               j = -10     3 9                 j = -12   1 5                   j = -14   3                     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, 169]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 169]]
Out[4]=	PD[X[8, 1, 9, 2], X[20, 11, 21, 12], X[10, 4, 11, 3], X[2, 17, 3, 18], > X[14, 5, 15, 6], X[6, 7, 1, 8], X[16, 10, 17, 9], X[18, 13, 19, 14], > X[22, 16, 7, 15], X[12, 19, 13, 20], X[4, 22, 5, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -11, 5, -6}, > {6, -1, 7, -3, 2, -10, 8, -5, 9, -7, 4, -8, 10, -2, 11, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 4 8 14 19 22 22 20 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 14 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 2 -18 -16 4 3 4 2 -6 2 3 2 4 6 - q + --- - q + q + --- - --- + --- - -- - q + -- - -- - q + q + 20 14 12 10 8 4 2 q q q q q q q 6 8 10 > 2 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 169]][a, z]
Out[8]=	3 5 3 1 2 a 2 a a z 3 2 z 3 3 3 -(---) + --- - ---- + -- - - + 2 a z - 2 a z - ---- + 3 a z + 2 a z - a z z z z a a 5 5 3 z 5 3 5 5 5 7 3 7 > 2 a z - -- + 3 a z + 3 a z - a z + a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 169]][a, z]
Out[9]=	3 5 2 1 2 a 2 a a 2 z 3 5 2 2 2 -a - --- - --- - ---- - -- + --- + 6 a z + 6 a z + 2 a z + 5 z + 14 a z + a z z z z a 3 3 4 2 6 2 z 6 z 3 3 3 7 3 4 > 13 a z + 4 a z + -- - ---- - 6 a z + 6 a z - 5 a z - 13 z + 3 a a 4 5 5 5 z 2 4 4 4 6 4 8 4 z 13 z 5 > ---- - 38 a z - 35 a z - 13 a z + 2 a z - -- + ----- - 3 a z - 2 3 a a a 6 3 5 5 5 7 5 6 4 z 2 6 4 6 > 35 a z - 8 a z + 10 a z + 18 z - ---- + 34 a z + 31 a z + 2 a 7 6 6 8 6 8 z 7 3 7 5 7 7 7 8 > 18 a z - a z - ---- + 11 a z + 38 a z + 15 a z - 4 a z - 10 z - a 2 8 4 8 6 8 9 3 9 5 9 2 10 > 9 a z - 6 a z - 7 a z - 8 a z - 15 a z - 7 a z - 3 a z - 4 10 > 3 a z
In[10]:=	Kh[L][q, t]
Out[10]=	9 1 3 1 5 3 9 5 10 12 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 9 12 11 11 11 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 6 t + 8 q t + 3 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 L11a169

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