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DrawMorseLink

PD Presentation:

X10,1,11,2 X16,11,17,12 X20,5,21,6 X12,4,13,3 X14,8,15,7 X6,14,7,13 X22,17,9,18 X18,21,19,22 X8,9,1,10 X4,19,5,20 X2,16,3,15

Gauss Code:

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

Jones Polynomial:

q-15/2 - 4q-13/2 + 9q-11/2 - 14q-9/2 + 18q-7/2 - 21q-5/2 + 20q-3/2 - 18q-1/2 + 12q1/2 - 7q3/2 + 3q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-22 + 2q-20 - 2q-18 + 2q-14 - 4q-12 + 4q-10 - q-8 + 2q-6 + 3q-4 - 2q-2 + 5 - 2q2 + q4 + 2q6 - q8 + q10

HOMFLY-PT Polynomial:

- a-1z-1 - 3a-1z - 3a-1z3 - a-1z5 + az-1 + 5az + 7az3 + 4az5 + az7 - a3z + 2a3z3 + 3a3z5 + a3z7 - a5z - 2a5z3 - a5z5

Kauffman Polynomial:

- a-3z + 2a-3z3 - a-3z5 - 2a-2z2 + 5a-2z4 - 3a-2z6 - a-1z-1 + 3a-1z - 4a-1z3 + 7a-1z5 - 5a-1z7 + 1 - 3z4 + 7z6 - 6z8 - az-1 + 8az - 15az3 + 8az5 + 2az7 - 5az9 + 3a2z2 - 21a2z4 + 25a2z6 - 9a2z8 - 2a2z10 + 4a3z - 9a3z3 - 4a3z5 + 19a3z7 - 11a3z9 + 5a4z2 - 24a4z4 + 32a4z6 - 10a4z8 - 2a4z10 + a5z - 5a5z3 + 5a5z5 + 8a5z7 - 6a5z9 + 3a6z2 - 9a6z4 + 16a6z6 - 7a6z8 + a7z - 5a7z3 + 9a7z5 - 4a7z7 - a8z2 + 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                     2   j = 4                   5 1   j = 2                 7 2     j = 0               11 5       j = -2             11 9         j = -4           10 9           j = -6         8 11             j = -8       6 10               j = -10     3 8                 j = -12   1 6                   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, 321]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 321]]
Out[4]=	PD[X[10, 1, 11, 2], X[16, 11, 17, 12], X[20, 5, 21, 6], X[12, 4, 13, 3], > X[14, 8, 15, 7], X[6, 14, 7, 13], X[22, 17, 9, 18], X[18, 21, 19, 22], > X[8, 9, 1, 10], X[4, 19, 5, 20], X[2, 16, 3, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 4, -10, 3, -6, 5, -9}, > {9, -1, 2, -4, 6, -5, 11, -2, 7, -8, 10, -3, 8, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 4 9 14 18 21 20 18 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 12 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 > 7 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 2 2 2 4 4 -8 2 3 2 2 4 5 - q + --- - --- + --- - --- + --- - q + -- + -- - -- - 2 q + q + 20 18 14 12 10 6 4 2 q q q q q q q q 6 8 10 > 2 q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 321]][a, z]
Out[8]=	3 1 a 3 z 3 5 3 z 3 3 3 5 3 -(---) + - - --- + 5 a z - a z - a z - ---- + 7 a z + 2 a z - 2 a z - a z z a a 5 z 5 3 5 5 5 7 3 7 > -- + 4 a z + 3 a z - a z + a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 321]][a, z]
Out[9]=	2 1 a z 3 z 3 5 7 2 z 2 2 1 - --- - - - -- + --- + 8 a z + 4 a z + a z + a z - ---- + 3 a z + a z z 3 a 2 a a 3 3 4 2 6 2 8 2 2 z 4 z 3 3 3 5 3 > 5 a z + 3 a z - a z + ---- - ---- - 15 a z - 9 a z - 5 a z - 3 a a 4 5 7 3 4 5 z 2 4 4 4 6 4 8 4 z > 5 a z - 3 z + ---- - 21 a z - 24 a z - 9 a z + 2 a z - -- + 2 3 a a 5 6 7 z 5 3 5 5 5 7 5 6 3 z 2 6 > ---- + 8 a z - 4 a z + 5 a z + 9 a z + 7 z - ---- + 25 a z + a 2 a 7 4 6 6 6 8 6 5 z 7 3 7 5 7 > 32 a z + 16 a z - a z - ---- + 2 a z + 19 a z + 8 a z - a 7 7 8 2 8 4 8 6 8 9 3 9 > 4 a z - 6 z - 9 a z - 10 a z - 7 a z - 5 a z - 11 a z - 5 9 2 10 4 10 > 6 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	9 1 3 1 6 3 8 6 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 8 11 10 9 11 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 5 t + 7 q t + 2 q t + 5 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 + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a321

L11a320 L11a322