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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X7,16,8,17 X22,18,5,17 X18,12,19,11 X20,14,21,13 X12,20,13,19 X14,22,15,21 X15,8,16,9 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

- 2q-7/2 + 3q-5/2 - 6q-3/2 + 7q-1/2 - 9q1/2 + 9q3/2 - 7q5/2 + 5q7/2 - 3q9/2 + q11/2

A2 (sl(3)) Invariant:

q-12 + 3q-10 + 2q-8 + 4q-6 + q-4 + 1 - 3q2 + q4 - 2q6 + q8 + q10 - q12 + q14 - q16

HOMFLY-PT Polynomial:

2a-3z + 3a-3z3 + a-3z5 + a-1z-1 - 3a-1z - 8a-1z3 - 5a-1z5 - a-1z7 - 3az-1 - 2az + 2az3 + az5 + 2a3z-1 + a3z

Kauffman Polynomial:

a-6z2 - a-6z4 + 4a-5z3 - 3a-5z5 - a-4z2 + 5a-4z4 - 4a-4z6 + 2a-3z - 5a-3z3 + 6a-3z5 - 4a-3z7 - a-2 + a-2z2 - 5a-2z4 + 5a-2z6 - 3a-2z8 + a-1z-1 + a-1z - 9a-1z3 + 8a-1z5 - 2a-1z7 - a-1z9 - 3 + 9z2 - 13z4 + 11z6 - 4z8 + 3az-1 - 9az + 10az3 - 4az5 + 2az7 - az9 - 3a2 + 6a2z2 - 2a2z4 + 2a2z6 - a2z8 + 2a3z-1 - 8a3z + 10a3z3 - 3a3z5

Khovanov Homology:

trqj 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                 2   j = 8               3 1   j = 6             4 2     j = 4           5 3       j = 2         4 4         j = 0       4 6           j = -2     2 3             j = -4   1 4               j = -6 1 2                 j = -8 2

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, 69]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 69]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[7, 16, 8, 17], X[22, 18, 5, 17], > X[18, 12, 19, 11], X[20, 14, 21, 13], X[12, 20, 13, 19], X[14, 22, 15, 21], > X[15, 8, 16, 9], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, 5, -7, 6, -8, -9, 3, 4, -5, > 7, -6, 8, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-2 3 6 7 3/2 5/2 7/2 9/2 ---- + ---- - ---- + ------- - 9 Sqrt[q] + 9 q - 7 q + 5 q - 3 q + 7/2 5/2 3/2 Sqrt[q] q q q 11/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 3 2 4 -4 2 4 6 8 10 12 14 16 1 + q + --- + -- + -- + q - 3 q + q - 2 q + q + q - q + q - q 10 8 6 q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 69]][a, z]
Out[8]=	3 3 3 5 1 3 a 2 a 2 z 3 z 3 3 z 8 z 3 z --- - --- + ---- + --- - --- - 2 a z + a z + ---- - ---- + 2 a z + -- - a z z z 3 a 3 a 3 a a a 5 7 5 z 5 z > ---- + a z - -- a a
In[9]:=	Kauffman[Link[11, NonAlternating, 69]][a, z]
Out[9]=	3 2 -2 2 1 3 a 2 a 2 z z 3 2 z -3 - a - 3 a + --- + --- + ---- + --- + - - 9 a z - 8 a z + 9 z + -- - a z z z 3 a 6 a a 2 2 3 3 3 4 z z 2 2 4 z 5 z 9 z 3 3 3 4 z > -- + -- + 6 a z + ---- - ---- - ---- + 10 a z + 10 a z - 13 z - -- + 4 2 5 3 a 6 a a a a a 4 4 5 5 5 5 z 5 z 2 4 3 z 6 z 8 z 5 3 5 6 > ---- - ---- - 2 a z - ---- + ---- + ---- - 4 a z - 3 a z + 11 z - 4 2 5 3 a a a a a 6 6 7 7 8 9 4 z 5 z 2 6 4 z 2 z 7 8 3 z 2 8 z > ---- + ---- + 2 a z - ---- - ---- + 2 a z - 4 z - ---- - a z - -- - 4 2 3 a 2 a a a a a 9 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 2 1 4 2 4 3 2 6 + 4 q + ----- + ----- + ----- + ----- + ----- + ----- + - + ---- + 4 q t + 8 4 6 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t q t 4 4 2 6 2 6 3 8 3 8 4 10 4 12 5 > 5 q t + 3 q t + 4 q t + 2 q t + 3 q t + q t + 2 q t + q t

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L11n68 L11n70