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DrawMorseLink

PD Presentation:

X6172 X10,4,11,3 X12,8,13,7 X18,14,19,13 X16,9,17,10 X8,17,9,18 X22,20,5,19 X20,15,21,16 X14,21,15,22 X2536 X4,12,1,11

Gauss Code:

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

Jones Polynomial:

- q-11/2 + 3q-9/2 - 7q-7/2 + 12q-5/2 - 17q-3/2 + 20q-1/2 - 21q1/2 + 18q3/2 - 15q5/2 + 9q7/2 - 4q9/2 + q11/2

A2 (sl(3)) Invariant:

q-18 + q-16 - q-14 + 2q-12 - 3q-8 + 3q-6 - 3q-4 + q-2 + 2 - q2 + 6q4 - 2q6 + 4q8 + q10 - 3q12 + 2q14 - q16

HOMFLY-PT Polynomial:

2a-3z + 2a-3z3 + a-3z5 - 2a-1z-1 - 7a-1z - 8a-1z3 - 4a-1z5 - a-1z7 + 4az-1 + 10az + 9az3 + 3az5 - 3a3z-1 - 6a3z - 3a3z3 + a5z-1 + a5z

Kauffman Polynomial:

- a-6z4 + a-5z3 - 4a-5z5 - 2a-4z2 + 7a-4z4 - 9a-4z6 + 5a-3z - 14a-3z3 + 21a-3z5 - 14a-3z7 - a-2 + a-2z2 - 5a-2z4 + 18a-2z6 - 13a-2z8 - 2a-1z-1 + 19a-1z - 51a-1z3 + 54a-1z5 - 11a-1z7 - 6a-1z9 - 2 + 16z2 - 50z4 + 63z6 - 22z8 - z10 - 4az-1 + 28az - 61az3 + 41az5 + 7az7 - 9az9 - 3a2 + 20a2z2 - 51a2z4 + 47a2z6 - 12a2z8 - a2z10 - 3a3z-1 + 18a3z - 31a3z3 + 16a3z5 + 3a3z7 - 3a3z9 - a4 + 7a4z2 - 14a4z4 + 11a4z6 - 3a4z8 - a5z-1 + 4a5z - 6a5z3 + 4a5z5 - a5z7

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 12                       1 j = 10                     3   j = 8                   6 1   j = 6                 9 3     j = 4               9 6       j = 2             12 9         j = 0           10 11           j = -2         7 10             j = -4       5 10               j = -6     2 7                 j = -8   1 5                   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, 18]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 18]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[12, 8, 13, 7], X[18, 14, 19, 13], > X[16, 9, 17, 10], X[8, 17, 9, 18], X[22, 20, 5, 19], X[20, 15, 21, 16], > X[14, 21, 15, 22], X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -6, 5, -2, 11, -3, 4, -9, 8, -5, 6, -4, > 7, -8, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 3 7 12 17 20 3/2 -q + ---- - ---- + ---- - ---- + ------- - 21 Sqrt[q] + 18 q - 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 11/2 > 15 q + 9 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 -14 2 3 3 3 -2 2 4 6 8 2 + q + q - q + --- - -- + -- - -- + q - q + 6 q - 2 q + 4 q + 12 8 6 4 q q q q 10 12 14 16 > q - 3 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 18]][a, z]
Out[8]=	3 5 3 3 -2 4 a 3 a a 2 z 7 z 3 5 2 z 8 z --- + --- - ---- + -- + --- - --- + 10 a z - 6 a z + a z + ---- - ---- + a z z z z 3 a 3 a a a 5 5 7 3 3 3 z 4 z 5 z > 9 a z - 3 a z + -- - ---- + 3 a z - -- 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 18]][a, z]
Out[9]=	3 5 -2 2 4 2 4 a 3 a a 5 z 19 z 3 -2 - a - 3 a - a - --- - --- - ---- - -- + --- + ---- + 28 a z + 18 a z + a z z z z 3 a a 2 2 3 3 3 5 2 2 z z 2 2 4 2 z 14 z 51 z > 4 a z + 16 z - ---- + -- + 20 a z + 7 a z + -- - ----- - ----- - 4 2 5 3 a a a a a 4 4 4 3 3 3 5 3 4 z 7 z 5 z 2 4 > 61 a z - 31 a z - 6 a z - 50 z - -- + ---- - ---- - 51 a z - 6 4 2 a a a 5 5 5 4 4 4 z 21 z 54 z 5 3 5 5 5 6 > 14 a z - ---- + ----- + ----- + 41 a z + 16 a z + 4 a z + 63 z - 5 3 a a a 6 6 7 7 9 z 18 z 2 6 4 6 14 z 11 z 7 3 7 > ---- + ----- + 47 a z + 11 a z - ----- - ----- + 7 a z + 3 a z - 4 2 3 a a a a 8 9 5 7 8 13 z 2 8 4 8 6 z 9 3 9 > a z - 22 z - ----- - 12 a z - 3 a z - ---- - 9 a z - 3 a z - 2 a a 10 2 10 > z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 5 2 7 5 10 11 + 12 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 12 6 10 5 8 5 8 4 6 4 6 3 4 3 4 2 q t q t q t q t q t q t q t q t 7 10 10 2 4 4 2 6 2 6 3 > ----- + -- + ---- + 9 q t + 9 q t + 6 q t + 9 q t + 3 q t + 2 2 t 2 q t q t 8 3 8 4 10 4 12 5 > 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a18

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