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DrawMorseLink

PD Presentation:

X8192 X12,4,13,3 X22,12,7,11 X20,16,21,15 X18,10,19,9 X10,20,11,19 X14,22,15,21 X16,6,17,5 X2738 X4,14,5,13 X6,18,1,17

Gauss Code:

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

Jones Polynomial:

- q1/2 + 3q3/2 - 8q5/2 + 11q7/2 - 16q9/2 + 18q11/2 - 18q13/2 + 16q15/2 - 12q17/2 + 7q19/2 - 3q21/2 + q23/2

A2 (sl(3)) Invariant:

q2 - q4 + 2q6 + 2q8 + 6q12 - q14 + 3q16 - 2q20 + 3q22 - 3q24 + 2q26 - q28 - 2q30 + q32 - q34

HOMFLY-PT Polynomial:

a-9z-1 + 3a-9z + 3a-9z3 + a-9z5 - 2a-7z-1 - 7a-7z - 7a-7z3 - 4a-7z5 - a-7z7 + 2a-5z - a-5z3 - 3a-5z5 - a-5z7 + a-3z-1 + 3a-3z + 3a-3z3 + a-3z5

Kauffman Polynomial:

a-14z2 - a-14z4 + 2a-13z3 - 3a-13z5 - 3a-12z2 + 6a-12z4 - 6a-12z6 + 2a-11z - 11a-11z3 + 14a-11z5 - 9a-11z7 - 2a-10 + 9a-10z2 - 19a-10z4 + 20a-10z6 - 10a-10z8 + a-9z-1 - a-9z - 2a-9z3 + 10a-9z7 - 7a-9z9 - 5a-8 + 27a-8z2 - 50a-8z4 + 45a-8z6 - 11a-8z8 - 2a-8z10 + 2a-7z-1 - 7a-7z + 11a-7z3 - 22a-7z5 + 30a-7z7 - 11a-7z9 - 3a-6 + 15a-6z2 - 33a-6z4 + 29a-6z6 - 4a-6z8 - 2a-6z10 - 6a-5z3 - a-5z5 + 10a-5z7 - 4a-5z9 + a-4 + a-4z2 - 9a-4z4 + 10a-4z6 - 3a-4z8 - a-3z-1 + 4a-3z - 6a-3z3 + 4a-3z5 - a-3z7

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 24                       1 j = 22                     2   j = 20                   5 1   j = 18                 7 2     j = 16               9 5       j = 14             9 7         j = 12           9 9           j = 10         7 9             j = 8       5 10               j = 6     3 6                 j = 4   1 6                   j = 2   2                     j = 0 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, 215]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 215]]
Out[4]=	PD[X[8, 1, 9, 2], X[12, 4, 13, 3], X[22, 12, 7, 11], X[20, 16, 21, 15], > X[18, 10, 19, 9], X[10, 20, 11, 19], X[14, 22, 15, 21], X[16, 6, 17, 5], > X[2, 7, 3, 8], X[4, 14, 5, 13], X[6, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10, 8, -11}, > {9, -1, 5, -6, 3, -2, 10, -7, 4, -8, 11, -5, 6, -4, 7, -3}]
In[6]:=	Jones[L][q]
Out[6]=	3/2 5/2 7/2 9/2 11/2 13/2 -Sqrt[q] + 3 q - 8 q + 11 q - 16 q + 18 q - 18 q + 15/2 17/2 19/2 21/2 23/2 > 16 q - 12 q + 7 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 4 6 8 12 14 16 20 22 24 26 q - q + 2 q + 2 q + 6 q - q + 3 q - 2 q + 3 q - 3 q + 2 q - 28 30 32 34 > q - 2 q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 215]][a, z]
Out[8]=	3 3 3 3 5 1 2 1 3 z 7 z 2 z 3 z 3 z 7 z z 3 z z ---- - ---- + ---- + --- - --- + --- + --- + ---- - ---- - -- + ---- + -- - 9 7 3 9 7 5 3 9 7 5 3 9 a z a z a z a a a a a a a a a 5 5 5 7 7 4 z 3 z z z z > ---- - ---- + -- - -- - -- 7 5 3 7 5 a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 215]][a, z]
Out[9]=	2 2 -2 5 3 -4 1 2 1 2 z z 7 z 4 z z 3 z --- - -- - -- + a + ---- + ---- - ---- + --- - -- - --- + --- + --- - ---- + 10 8 6 9 7 3 11 9 7 3 14 12 a a a a z a z a z a a a a a a 2 2 2 2 3 3 3 3 3 3 9 z 27 z 15 z z 2 z 11 z 2 z 11 z 6 z 6 z > ---- + ----- + ----- + -- + ---- - ----- - ---- + ----- - ---- - ---- - 10 8 6 4 13 11 9 7 5 3 a a a a a a a a a a 4 4 4 4 4 4 5 5 5 5 z 6 z 19 z 50 z 33 z 9 z 3 z 14 z 22 z z > --- + ---- - ----- - ----- - ----- - ---- - ---- + ----- - ----- - -- + 14 12 10 8 6 4 13 11 7 5 a a a a a a a a a a 5 6 6 6 6 6 7 7 7 4 z 6 z 20 z 45 z 29 z 10 z 9 z 10 z 30 z > ---- - ---- + ----- + ----- + ----- + ----- - ---- + ----- + ----- + 3 12 10 8 6 4 11 9 7 a a a a a a a a a 7 7 8 8 8 8 9 9 9 10 10 z z 10 z 11 z 4 z 3 z 7 z 11 z 4 z 2 z > ----- - -- - ----- - ----- - ---- - ---- - ---- - ----- - ---- - ----- - 5 3 10 8 6 4 9 7 5 8 a a a a a a a a a a 10 2 z > ----- 6 a
In[10]:=	Kh[L][q, t]
Out[10]=	2 4 4 6 -2 2 q q 6 8 8 2 10 2 6 q + 3 q + t + ---- + -- + 6 q t + 5 q t + 10 q t + 7 q t + t t 10 3 12 3 12 4 14 4 14 5 16 5 > 9 q t + 9 q t + 9 q t + 9 q t + 7 q t + 9 q t + 16 6 18 6 18 7 20 7 20 8 22 8 24 9 > 5 q t + 7 q t + 2 q t + 5 q t + q t + 2 q t + q t

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