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DrawMorseLink

PD Presentation:

X6172 X14,4,15,3 X16,8,17,7 X18,10,19,9 X22,14,5,13 X8,18,9,17 X10,22,11,21 X20,12,21,11 X12,20,13,19 X2536 X4,16,1,15

Gauss Code:

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

Jones Polynomial:

- q1/2 + 2q3/2 - 5q5/2 + 6q7/2 - 9q9/2 + 9q11/2 - 10q13/2 + 9q15/2 - 6q17/2 + 4q19/2 - 2q21/2 + q23/2

A2 (sl(3)) Invariant:

q2 + q6 + 2q8 + q10 + 4q12 + q14 + 4q16 + 2q18 + q22 - 3q24 - q26 - 2q28 - q30 - q34

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

2a-14z2 - a-14z4 + 3a-13z3 - 2a-13z5 + 2a-12 - 6a-12z2 + 6a-12z4 - 3a-12z6 - a-11z - 4a-11z3 + 5a-11z5 - 3a-11z7 + a-10 - 4a-10z2 - 4a-10z4 + 6a-10z6 - 3a-10z8 + 2a-9z-1 - 10a-9z + 22a-9z3 - 22a-9z5 + 11a-9z7 - 3a-9z9 - 6a-8 + 20a-8z2 - 22a-8z4 + 12a-8z6 - a-8z8 - a-8z10 + 4a-7z-1 - 17a-7z + 35a-7z3 - 35a-7z5 + 21a-7z7 - 5a-7z9 - 5a-6 + 15a-6z2 - 18a-6z4 + 11a-6z6 - a-6z10 + a-5z-1 - 3a-5z - 2a-5z3 - a-5z5 + 6a-5z7 - 2a-5z9 + a-4 - a-4z2 - 7a-4z4 + 8a-4z6 - 2a-4z8 - a-3z-1 + 5a-3z - 8a-3z3 + 5a-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                     1   j = 20                   3 1   j = 18                 3 1     j = 16               6 3       j = 14             4 3         j = 12           5 6           j = 10         4 4             j = 8       2 5               j = 6     3 4                 j = 4   1 4                   j = 2   1                     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, 111]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 111]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 4, 15, 3], X[16, 8, 17, 7], X[18, 10, 19, 9], > X[22, 14, 5, 13], X[8, 18, 9, 17], X[10, 22, 11, 21], X[20, 12, 21, 11], > X[12, 20, 13, 19], X[2, 5, 3, 6], X[4, 16, 1, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -6, 4, -7, 8, -9, 5, -2, 11, -3, 6, -4, > 9, -8, 7, -5}]
In[6]:=	Jones[L][q]
Out[6]=	3/2 5/2 7/2 9/2 11/2 13/2 15/2 -Sqrt[q] + 2 q - 5 q + 6 q - 9 q + 9 q - 10 q + 9 q - 17/2 19/2 21/2 23/2 > 6 q + 4 q - 2 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 6 8 10 12 14 16 18 22 24 26 q + q + 2 q + q + 4 q + q + 4 q + 2 q + q - 3 q - q - 28 30 34 > 2 q - q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 111]][a, z]
Out[8]=	3 3 3 3 2 4 1 1 4 z 9 z z 4 z 4 z 9 z 3 z 4 z ---- - ---- + ---- + ---- + --- - --- + -- + --- + ---- - ---- - ---- + ---- + 9 7 5 3 9 7 5 3 9 7 5 3 a z a z a z a z a a a a a a a a 5 5 5 5 7 7 z 5 z 4 z z z z > -- - ---- - ---- + -- - -- - -- 9 7 5 3 7 5 a a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 111]][a, z]
Out[9]=	2 -10 6 5 -4 2 4 1 1 z 10 z 17 z --- + a - -- - -- + a + ---- + ---- + ---- - ---- - --- - ---- - ---- - 12 8 6 9 7 5 3 11 9 7 a a a a z a z a z a z a a a 2 2 2 2 2 2 3 3 3 3 z 5 z 2 z 6 z 4 z 20 z 15 z z 3 z 4 z 22 z > --- + --- + ---- - ---- - ---- + ----- + ----- - -- + ---- - ---- + ----- + 5 3 14 12 10 8 6 4 13 11 9 a a a a a a a a a a a 3 3 3 4 4 4 4 4 4 5 35 z 2 z 8 z z 6 z 4 z 22 z 18 z 7 z 2 z > ----- - ---- - ---- - --- + ---- - ---- - ----- - ----- - ---- - ---- + 7 5 3 14 12 10 8 6 4 13 a a a a a a a a a a 5 5 5 5 5 6 6 6 6 6 5 z 22 z 35 z z 5 z 3 z 6 z 12 z 11 z 8 z > ---- - ----- - ----- - -- + ---- - ---- + ---- + ----- + ----- + ---- - 11 9 7 5 3 12 10 8 6 4 a a a a a a a a a a 7 7 7 7 7 8 8 8 9 9 9 3 z 11 z 21 z 6 z z 3 z z 2 z 3 z 5 z 2 z > ---- + ----- + ----- + ---- - -- - ---- - -- - ---- - ---- - ---- - ---- - 11 9 7 5 3 10 8 4 9 7 5 a a a a a a a a a a a 10 10 z z > --- - --- 8 6 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 4 4 6 -2 q q 6 8 8 2 10 2 10 3 4 q + 3 q + t + -- + -- + 4 q t + 2 q t + 5 q t + 4 q t + 4 q t + t t 12 3 12 4 14 4 14 5 16 5 16 6 > 5 q t + 6 q t + 4 q t + 3 q t + 6 q t + 3 q t + 18 6 18 7 20 7 20 8 22 8 24 9 > 3 q t + q t + 3 q t + q t + q t + q t

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