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DrawMorseLink

PD Presentation:

X8192 X20,9,21,10 X4758 X16,5,17,6 X14,18,15,17 X6,15,1,16 X22,14,7,13 X18,22,19,21 X2,11,3,12 X12,3,13,4 X10,19,11,20

Gauss Code:

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

Jones Polynomial:

- q-19/2 + 3q-17/2 - 8q-15/2 + 14q-13/2 - 20q-11/2 + 23q-9/2 - 24q-7/2 + 21q-5/2 - 16q-3/2 + 10q-1/2 - 5q1/2 + q3/2

A2 (sl(3)) Invariant:

q-30 + q-28 - q-26 + 3q-24 + q-22 - 3q-20 + 4q-18 - 3q-16 + 2q-14 + 2q-12 - 2q-10 + 5q-8 - 5q-6 + 3q-4 + q-2 - 2 + 3q2 - q4

HOMFLY-PT Polynomial:

- az + az3 + az5 - a3z-1 - 3a3z - 4a3z3 - 3a3z5 - a3z7 + 2a5z-1 + 5a5z + 7a5z3 + 3a5z5 - 2a7z-1 - 5a7z - 3a7z3 + a9z-1 + a9z

Kauffman Polynomial:

z4 - z6 - az - 4az3 + 10az5 - 5az7 + a2z2 - 11a2z4 + 20a2z6 - 9a2z8 - a3z-1 + 5a3z - 14a3z3 + 16a3z5 + 5a3z7 - 7a3z9 + 8a4z2 - 34a4z4 + 48a4z6 - 17a4z8 - 2a4z10 - 2a5z-1 + 12a5z - 26a5z3 + 15a5z5 + 15a5z7 - 13a5z9 - a6 + 12a6z2 - 37a6z4 + 42a6z6 - 16a6z8 - 2a6z10 - 2a7z-1 + 12a7z - 26a7z3 + 19a7z5 - a7z7 - 6a7z9 + 4a8z2 - 11a8z4 + 12a8z6 - 8a8z8 - a9z-1 + 5a9z - 8a9z3 + 9a9z5 - 6a9z7 - a10z2 + 4a10z4 - 3a10z6 - a11z + 2a11z3 - a11z5

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 4                       1 j = 2                     4   j = 0                   6 1   j = -2                 10 4     j = -4               12 7       j = -6             12 9         j = -8           11 12           j = -10         9 12             j = -12       5 11               j = -14     3 9                 j = -16   1 6                   j = -18   2                     j = -20 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, 227]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 227]]
Out[4]=	PD[X[8, 1, 9, 2], X[20, 9, 21, 10], X[4, 7, 5, 8], X[16, 5, 17, 6], > X[14, 18, 15, 17], X[6, 15, 1, 16], X[22, 14, 7, 13], X[18, 22, 19, 21], > X[2, 11, 3, 12], X[12, 3, 13, 4], X[10, 19, 11, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 10, -3, 4, -6}, > {3, -1, 2, -11, 9, -10, 7, -5, 6, -4, 5, -8, 11, -2, 8, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 3 8 14 20 23 24 21 16 -q + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q 10 3/2 > ------- - 5 Sqrt[q] + q Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 -28 -26 3 -22 3 4 3 2 2 2 5 -2 + q + q - q + --- + q - --- + --- - --- + --- + --- - --- + -- - 24 20 18 16 14 12 10 8 q q q q q q q q 5 3 -2 2 4 > -- + -- + q + 3 q - q 6 4 q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 227]][a, z]
Out[8]=	3 5 7 9 a 2 a 2 a a 3 5 7 9 3 -(--) + ---- - ---- + -- - a z - 3 a z + 5 a z - 5 a z + a z + a z - z z z z 3 3 5 3 7 3 5 3 5 5 5 3 7 > 4 a z + 7 a z - 3 a z + a z - 3 a z + 3 a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 227]][a, z]
Out[9]=	3 5 7 9 6 a 2 a 2 a a 3 5 7 9 -a - -- - ---- - ---- - -- - a z + 5 a z + 12 a z + 12 a z + 5 a z - z z z z 11 2 2 4 2 6 2 8 2 10 2 3 3 3 > a z + a z + 8 a z + 12 a z + 4 a z - a z - 4 a z - 14 a z - 5 3 7 3 9 3 11 3 4 2 4 4 4 > 26 a z - 26 a z - 8 a z + 2 a z + z - 11 a z - 34 a z - 6 4 8 4 10 4 5 3 5 5 5 7 5 > 37 a z - 11 a z + 4 a z + 10 a z + 16 a z + 15 a z + 19 a z + 9 5 11 5 6 2 6 4 6 6 6 8 6 > 9 a z - a z - z + 20 a z + 48 a z + 42 a z + 12 a z - 10 6 7 3 7 5 7 7 7 9 7 2 8 > 3 a z - 5 a z + 5 a z + 15 a z - a z - 6 a z - 9 a z - 4 8 6 8 8 8 3 9 5 9 7 9 4 10 > 17 a z - 16 a z - 8 a z - 7 a z - 13 a z - 6 a z - 2 a z - 6 10 > 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	7 10 1 2 1 6 3 9 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 20 8 18 7 16 7 16 6 14 6 14 5 12 5 q q q t q t q t q t q t q t q t 11 9 12 11 12 12 9 12 > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + 6 t + 12 4 10 4 10 3 8 3 8 2 6 2 6 4 q t q t q t q t q t q t q t q t 4 t 2 2 2 4 3 > --- + t + 4 q t + q t 2 q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a227

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