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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X14,6,15,5 X8493 X18,10,19,9 X20,12,21,11 X22,14,5,13 X10,20,11,19 X12,22,13,21 X2,16,3,15

Gauss Code:

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

Jones Polynomial:

q-1/2 - 3q1/2 + 3q3/2 - 6q5/2 + 6q7/2 - 8q9/2 + 8q11/2 - 7q13/2 + 6q15/2 - 4q17/2 + 3q19/2 - q21/2

A2 (sl(3)) Invariant:

- 1 + q2 + q4 + 3q6 + 4q8 + 3q10 + 4q12 + 2q16 - 2q18 - 2q20 - 2q22 - 2q24 - q28 + q30

HOMFLY-PT Polynomial:

2a-7z-1 - 6a-7z3 - 5a-7z5 - a-7z7 - 5a-5z-1 - 3a-5z + 12a-5z3 + 16a-5z5 + 7a-5z7 + a-5z9 + 3a-3z-1 + 3a-3z - 5a-3z3 - 5a-3z5 - a-3z7

Kauffman Polynomial:

- a-13z3 + 2a-12z2 - 3a-12z4 + 3a-11z3 - 4a-11z5 - a-10 + 2a-10z2 + 3a-10z4 - 4a-10z6 + 6a-9z5 - 4a-9z7 - 4a-8z4 + 10a-8z6 - 4a-8z8 - 2a-7z-1 - a-7z + 12a-7z3 - 20a-7z5 + 16a-7z7 - 4a-7z9 + 5a-6 - 4a-6z2 - 6a-6z4 + a-6z6 + 6a-6z8 - 2a-6z10 - 5a-5z-1 + a-5z + 35a-5z3 - 63a-5z5 + 38a-5z7 - 7a-5z9 + 5a-4 - 3a-4z2 - 2a-4z4 - 8a-4z6 + 9a-4z8 - 2a-4z10 - 3a-3z-1 + 2a-3z + 19a-3z3 - 33a-3z5 + 18a-3z7 - 3a-3z9 + a-2z2 - 6a-2z4 + 5a-2z6 - a-2z8

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 22                       1 j = 20                     2   j = 18                   2 1   j = 16                 4 2     j = 14               3 2       j = 12             5 4         j = 10           3 3           j = 8         3 5             j = 6       3 3               j = 4     2 5                 j = 2   1 1                   j = 0   2                     j = -2 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, 16]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 16]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[14, 6, 15, 5], > X[8, 4, 9, 3], X[18, 10, 19, 9], X[20, 12, 21, 11], X[22, 14, 5, 13], > X[10, 20, 11, 19], X[12, 22, 13, 21], X[2, 16, 3, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {4, -1, 2, -5, 6, -9, 7, -10, 8, -4, 11, -2, 3, -6, > 9, -7, 10, -8}]
In[6]:=	Jones[L][q]
Out[6]=	1 3/2 5/2 7/2 9/2 11/2 13/2 ------- - 3 Sqrt[q] + 3 q - 6 q + 6 q - 8 q + 8 q - 7 q + Sqrt[q] 15/2 17/2 19/2 21/2 > 6 q - 4 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 4 6 8 10 12 16 18 20 22 -1 + q + q + 3 q + 4 q + 3 q + 4 q + 2 q - 2 q - 2 q - 2 q - 24 28 30 > 2 q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 16]][a, z]
Out[8]=	3 3 3 5 5 5 2 5 3 3 z 3 z 6 z 12 z 5 z 5 z 16 z 5 z ---- - ---- + ---- - --- + --- - ---- + ----- - ---- - ---- + ----- - ---- - 7 5 3 5 3 7 5 3 7 5 3 a z a z a z a a a a a a a a 7 7 7 9 z 7 z z z > -- + ---- - -- + -- 7 5 3 5 a a a a
In[9]:=	Kauffman[Link[11, Alternating, 16]][a, z]
Out[9]=	2 2 2 -10 5 5 2 5 3 z z 2 z 2 z 2 z 4 z -a + -- + -- - ---- - ---- - ---- - -- + -- + --- + ---- + ---- - ---- - 6 4 7 5 3 7 5 3 12 10 6 a a a z a z a z a a a a a a 2 2 3 3 3 3 3 4 4 4 3 z z z 3 z 12 z 35 z 19 z 3 z 3 z 4 z > ---- + -- - --- + ---- + ----- + ----- + ----- - ---- + ---- - ---- - 4 2 13 11 7 5 3 12 10 8 a a a a a a a a a a 4 4 4 5 5 5 5 5 6 6 6 z 2 z 6 z 4 z 6 z 20 z 63 z 33 z 4 z 10 z > ---- - ---- - ---- - ---- + ---- - ----- - ----- - ----- - ---- + ----- + 6 4 2 11 9 7 5 3 10 8 a a a a a a a a a a 6 6 6 7 7 7 7 8 8 8 8 z 8 z 5 z 4 z 16 z 38 z 18 z 4 z 6 z 9 z z > -- - ---- + ---- - ---- + ----- + ----- + ----- - ---- + ---- + ---- - -- - 6 4 2 9 7 5 3 8 6 4 2 a a a a a a a a a a a 9 9 9 10 10 4 z 7 z 3 z 2 z 2 z > ---- - ---- - ---- - ----- - ----- 7 5 3 6 4 a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 4 6 1 2 q q 2 q 6 8 8 2 5 q + 3 q + ----- + -- + -- + -- + ---- + 3 q t + 3 q t + 5 q t + 2 3 2 2 t t q t t t 10 2 10 3 12 3 12 4 14 4 14 5 > 3 q t + 3 q t + 5 q t + 4 q t + 3 q t + 2 q t + 16 5 16 6 18 6 18 7 20 7 22 8 > 4 q t + 2 q t + 2 q t + q t + 2 q t + q t

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