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DrawMorseLink

PD Presentation:

X8192 X9,21,10,20 X21,1,22,6 X18,8,19,7 X3,10,4,11 X15,12,16,13 X5,14,6,15 X13,4,14,5 X11,16,12,17 X22,18,7,17 X2,20,3,19

Gauss Code:

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

Jones Polynomial:

- q-9/2 + 3q-7/2 - 5q-5/2 + 6q-3/2 - 8q-1/2 + 6q1/2 - 6q3/2 + 4q5/2 - 2q7/2 + q9/2

A2 (sl(3)) Invariant:

q-14 - q-12 + q-8 - q-6 + 3q-4 + 2q-2 + 4 + 3q2 + q6 - 2q8 - q10 - q14

HOMFLY-PT Polynomial:

a-3z-1 + 2a-3z + a-3z3 - 3a-1z-1 - 4a-1z - 3a-1z3 - a-1z5 + 2az-1 - 2az3 - az5 + a3z + a3z3

Kauffman Polynomial:

a-4 - 4a-4z2 + 4a-4z4 - a-4z6 - a-3z-1 + 3a-3z - 6a-3z3 + 7a-3z5 - 2a-3z7 + 3a-2 - 5a-2z2 + a-2z4 + 5a-2z6 - 2a-2z8 - 3a-1z-1 + 7a-1z - 10a-1z3 + 8a-1z5 - a-1z9 + 3 + 2z2 - 13z4 + 13z6 - 4z8 - 2az-1 + 4az - 6az3 + 2az5 + az7 - az9 + 6a2z2 - 13a2z4 + 7a2z6 - 2a2z8 + a3z - 3a3z3 + a3z5 - a3z7 + 3a4z2 - 3a4z4 + a5z - a5z3

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 10                   1 j = 8                 1   j = 6               3 1   j = 4             3 1     j = 2           3 3       j = 0         5 3         j = -2       2 4           j = -4     3 4             j = -6   1 3               j = -8   2                 j = -10 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, NonAlternating, 185]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 185]]
Out[4]=	PD[X[8, 1, 9, 2], X[9, 21, 10, 20], X[21, 1, 22, 6], X[18, 8, 19, 7], > X[3, 10, 4, 11], X[15, 12, 16, 13], X[5, 14, 6, 15], X[13, 4, 14, 5], > X[11, 16, 12, 17], X[22, 18, 7, 17], X[2, 20, 3, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, -5, 8, -7, 3}, > {4, -1, -2, 5, -9, 6, -8, 7, -6, 9, 10, -4, 11, 2, -3, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 3 5 6 8 3/2 5/2 -q + ---- - ---- + ---- - ------- + 6 Sqrt[q] - 6 q + 4 q - 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 > 2 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -12 -8 -6 3 2 2 6 8 10 14 4 + q - q + q - q + -- + -- + 3 q + q - 2 q - q - q 4 2 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 185]][a, z]
Out[8]=	3 3 5 1 3 2 a 2 z 4 z 3 z 3 z 3 3 3 z 5 ---- - --- + --- + --- - --- + a z + -- - ---- - 2 a z + a z - -- - a z 3 a z z 3 a 3 a a a z a a
In[9]:=	Kauffman[Link[11, NonAlternating, 185]][a, z]
Out[9]=	-4 3 1 3 2 a 3 z 7 z 3 5 2 3 + a + -- - ---- - --- - --- + --- + --- + 4 a z + a z + a z + 2 z - 2 3 a z z 3 a a a z a 2 2 3 3 4 z 5 z 2 2 4 2 6 z 10 z 3 3 3 5 3 > ---- - ---- + 6 a z + 3 a z - ---- - ----- - 6 a z - 3 a z - a z - 4 2 3 a a a a 4 4 5 5 4 4 z z 2 4 4 4 7 z 8 z 5 3 5 > 13 z + ---- + -- - 13 a z - 3 a z + ---- + ---- + 2 a z + a z + 4 2 3 a a a a 6 6 7 8 6 z 5 z 2 6 2 z 7 3 7 8 2 z 2 8 > 13 z - -- + ---- + 7 a z - ---- + a z - a z - 4 z - ---- - 2 a z - 4 2 3 2 a a a a 9 z 9 > -- - a z a
In[10]:=	Kh[L][q, t]
Out[10]=	4 1 2 1 3 3 4 2 2 5 + -- + ------ + ----- + ----- + ----- + ----- + ---- + ---- + 3 t + 3 q t + 2 10 4 8 3 6 3 6 2 4 2 4 2 q q t q t q t q t q t q t q t 2 2 4 2 4 3 6 3 6 4 8 4 10 5 > 3 q t + 3 q t + q t + 3 q t + q t + q t + q t

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