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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-11/2 - 3q-9/2 + 6q-7/2 - 10q-5/2 + 13q-3/2 - 16q-1/2 + 14q1/2 - 13q3/2 + 10q5/2 - 6q7/2 + 3q9/2 - q11/2

A2 (sl(3)) Invariant:

- q-16 + q-14 - q-12 + 2q-8 - 2q-6 + 4q-4 + q-2 + 3 + 3q2 - 2q4 + 2q6 - 2q8 + q12 - q14 + q16

HOMFLY-PT Polynomial:

- 2a-3z - 3a-3z3 - a-3z5 - a-1z-1 + 2a-1z + 5a-1z3 + 4a-1z5 + a-1z7 + az-1 + 2az + 5az3 + 4az5 + az7 - 2a3z - 3a3z3 - a3z5

Kauffman Polynomial:

- 4a-5z3 + 4a-5z5 - a-5z7 + 5a-4z2 - 14a-4z4 + 12a-4z6 - 3a-4z8 - 4a-3z + 12a-3z3 - 18a-3z5 + 15a-3z7 - 4a-3z9 + 7a-2z2 - 16a-2z4 + 13a-2z6 + a-2z8 - 2a-2z10 - a-1z-1 - 4a-1z + 19a-1z3 - 33a-1z5 + 30a-1z7 - 9a-1z9 + 1 + 4z2 - 17z4 + 19z6 - 3z8 - 2z10 - az-1 + 4az - 11az3 + 4az5 + 7az7 - 5az9 - 9a2z4 + 13a2z6 - 7a2z8 + 4a3z - 11a3z3 + 12a3z5 - 7a3z7 - a4z2 + 5a4z4 - 5a4z6 + 3a5z3 - 3a5z5 + a6z2 - a6z4

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 12                       1 j = 10                     2   j = 8                   4 1   j = 6                 6 2     j = 4               7 4       j = 2             7 6         j = 0           9 7           j = -2         6 9             j = -4       4 7               j = -6     2 6                 j = -8   1 4                   j = -10   2                     j = -12 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, 313]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 313]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[18, 6, 19, 5], > X[14, 8, 15, 7], X[20, 13, 21, 14], X[8, 21, 1, 22], X[16, 10, 17, 9], > X[22, 16, 9, 15], X[6, 18, 7, 17], X[4, 20, 5, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -11, 4, -10, 5, -7}, > {8, -1, 2, -3, 6, -5, 9, -8, 10, -4, 11, -6, 7, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 3 6 10 13 16 3/2 q - ---- + ---- - ---- + ---- - ------- + 14 Sqrt[q] - 13 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 11/2 > 10 q - 6 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -14 -12 2 2 4 -2 2 4 6 8 12 3 - q + q - q + -- - -- + -- + q + 3 q - 2 q + 2 q - 2 q + q - 8 6 4 q q q 14 16 > q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 313]][a, z]
Out[8]=	3 3 5 1 a 2 z 2 z 3 3 z 5 z 3 3 3 z -(---) + - - --- + --- + 2 a z - 2 a z - ---- + ---- + 5 a z - 3 a z - -- + a z z 3 a 3 a 3 a a a 5 7 4 z 5 3 5 z 7 > ---- + 4 a z - a z + -- + a z a a
In[9]:=	Kauffman[Link[11, Alternating, 313]][a, z]
Out[9]=	2 2 1 a 4 z 4 z 3 2 5 z 7 z 4 2 6 2 1 - --- - - - --- - --- + 4 a z + 4 a z + 4 z + ---- + ---- - a z + a z - a z z 3 a 4 2 a a a 3 3 3 4 4 z 12 z 19 z 3 3 3 5 3 4 14 z > ---- + ----- + ----- - 11 a z - 11 a z + 3 a z - 17 z - ----- - 5 3 a 4 a a a 4 5 5 5 16 z 2 4 4 4 6 4 4 z 18 z 33 z 5 > ----- - 9 a z + 5 a z - a z + ---- - ----- - ----- + 4 a z + 2 5 3 a a a a 6 6 7 3 5 5 5 6 12 z 13 z 2 6 4 6 z > 12 a z - 3 a z + 19 z + ----- + ----- + 13 a z - 5 a z - -- + 4 2 5 a a a 7 7 8 8 9 15 z 30 z 7 3 7 8 3 z z 2 8 4 z > ----- + ----- + 7 a z - 7 a z - 3 z - ---- + -- - 7 a z - ---- - 3 a 4 2 3 a a a a 9 10 9 z 9 10 2 z > ---- - 5 a z - 2 z - ----- a 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	9 1 2 1 4 2 6 4 7 9 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 2 12 5 10 4 8 4 8 3 6 3 6 2 4 2 4 q q t q t q t q t q t q t q t q t 6 2 2 2 4 2 4 3 6 3 6 4 > ---- + 7 t + 7 q t + 6 q t + 7 q t + 4 q t + 6 q t + 2 q t + 2 q t 8 4 8 5 10 5 12 6 > 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a313

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