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DrawMorseLink

PD Presentation:

X6172 X10,4,11,3 X12,8,13,7 X13,18,14,19 X9,17,10,16 X17,9,18,8 X19,22,20,5 X15,21,16,20 X21,15,22,14 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-3/2 + 2q-1/2 - 6q1/2 + 7q3/2 - 8q5/2 + 8q7/2 - 7q9/2 + 5q11/2 - 3q13/2 + q15/2

A2 (sl(3)) Invariant:

q-6 + q-4 + 4 + q2 + q4 + 2q6 - q8 + q10 - 2q12 + q16 - q18 + 2q20 - q24

HOMFLY-PT Polynomial:

a-7z - 2a-5z - 2a-5z3 + 2a-3z + 2a-3z3 + a-3z5 - a-1z-1 - 2a-1z - 2a-1z3 + az-1 + az

Kauffman Polynomial:

- 2a-8z2 + 3a-8z4 - a-8z6 + 2a-7z - 8a-7z3 + 10a-7z5 - 3a-7z7 + a-6z2 - 3a-6z4 + 8a-6z6 - 3a-6z8 + 4a-5z - 15a-5z3 + 17a-5z5 - 3a-5z7 - a-5z9 + 5a-4z2 - 14a-4z4 + 15a-4z6 - 5a-4z8 + 4a-3z - 13a-3z3 + 9a-3z5 - a-3z7 - a-3z9 + 3a-2z2 - 10a-2z4 + 6a-2z6 - 2a-2z8 - a-1z-1 + 4a-1z - 7a-1z3 + 2a-1z5 - a-1z7 + 1 + z2 - 2z4 - az-1 + 2az - az3

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 16                   1 j = 14                 2   j = 12               3 1   j = 10             4 2     j = 8           4 3       j = 6         4 4         j = 4       3 4           j = 2     3 4             j = 0   1 5               j = -2   1                 j = -4 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, 50]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 50]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[12, 8, 13, 7], X[13, 18, 14, 19], > X[9, 17, 10, 16], X[17, 9, 18, 8], X[19, 22, 20, 5], X[15, 21, 16, 20], > X[21, 15, 22, 14], 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]=	-(3/2) 2 3/2 5/2 7/2 9/2 11/2 -q + ------- - 6 Sqrt[q] + 7 q - 8 q + 8 q - 7 q + 5 q - Sqrt[q] 13/2 15/2 > 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 -4 2 4 6 8 10 12 16 18 20 24 4 + q + q + q + q + 2 q - q + q - 2 q + q - q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 50]][a, z]
Out[8]=	3 3 3 5 1 a z 2 z 2 z 2 z 2 z 2 z 2 z z -(---) + - + -- - --- + --- - --- + a z - ---- + ---- - ---- + -- a z z 7 5 3 a 5 3 a 3 a a a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 50]][a, z]
Out[9]=	2 2 2 2 1 a 2 z 4 z 4 z 4 z 2 2 z z 5 z 3 z 1 - --- - - + --- + --- + --- + --- + 2 a z + z - ---- + -- + ---- + ---- - a z z 7 5 3 a 8 6 4 2 a a a a a a a 3 3 3 3 4 4 4 4 8 z 15 z 13 z 7 z 3 4 3 z 3 z 14 z 10 z > ---- - ----- - ----- - ---- - a z - 2 z + ---- - ---- - ----- - ----- + 7 5 3 a 8 6 4 2 a a a a a a a 5 5 5 5 6 6 6 6 7 7 7 10 z 17 z 9 z 2 z z 8 z 15 z 6 z 3 z 3 z z > ----- + ----- + ---- + ---- - -- + ---- + ----- + ---- - ---- - ---- - -- - 7 5 3 a 8 6 4 2 7 5 3 a a a a a a a a a a 7 8 8 8 9 9 z 3 z 5 z 2 z z z > -- - ---- - ---- - ---- - -- - -- a 6 4 2 5 3 a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 2 4 4 2 6 2 6 3 5 + 3 q + ----- + - + ---- + 4 q t + 3 q t + 4 q t + 4 q t + 4 q t + 4 2 t 2 q t q t 8 3 8 4 10 4 10 5 12 5 12 6 14 6 > 4 q t + 3 q t + 4 q t + 2 q t + 3 q t + q t + 2 q t + 16 7 > q t

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L11n49 L11n51