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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q1/2 + 2q3/2 - 6q5/2 + 9q7/2 - 15q9/2 + 17q11/2 - 18q13/2 + 17q15/2 - 13q17/2 + 9q19/2 - 4q21/2 + q23/2

A2 (sl(3)) Invariant:

q2 + 2q6 + 3q8 + 6q12 + 3q16 + 2q18 - 3q20 + 2q22 - 5q24 - q28 - 2q30 + 2q32 - q34

HOMFLY-PT Polynomial:

a-9z-1 + a-9z + 2a-9z3 + a-9z5 - a-7z-1 - a-7z - 2a-7z3 - 3a-7z5 - a-7z7 - 2a-5z-1 - 5a-5z - 6a-5z3 - 4a-5z5 - a-5z7 + 2a-3z-1 + 5a-3z + 4a-3z3 + a-3z5

Kauffman Polynomial:

- a-14z4 + a-13z3 - 4a-13z5 + a-12 - 4a-12z2 + 9a-12z4 - 9a-12z6 + a-11z - 7a-11z3 + 16a-11z5 - 12a-11z7 - 4a-10z2 + 4a-10z4 + 10a-10z6 - 10a-10z8 + a-9z-1 + 2a-9z - 12a-9z3 + 22a-9z5 - 4a-9z7 - 5a-9z9 - 3a-8 + 6a-8z2 - 19a-8z4 + 33a-8z6 - 13a-8z8 - a-8z10 + a-7z-1 + 4a-7z - 15a-7z3 + 8a-7z5 + 11a-7z7 - 7a-7z9 + 4a-6z2 - 19a-6z4 + 21a-6z6 - 5a-6z8 - a-6z10 - 2a-5z-1 + 10a-5z - 20a-5z3 + 11a-5z5 + 2a-5z7 - 2a-5z9 + 3a-4 - 2a-4z2 - 6a-4z4 + 7a-4z6 - 2a-4z8 - 2a-3z-1 + 7a-3z - 9a-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                     3   j = 20                   6 1   j = 18                 7 3     j = 16               10 6       j = 14             8 7         j = 12           9 10           j = 10         6 8             j = 8       3 9               j = 6     3 6                 j = 4   1 5                   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, 21]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 21]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[12, 8, 13, 7], X[18, 10, 19, 9], > X[20, 14, 21, 13], X[22, 16, 5, 15], X[14, 22, 15, 21], X[16, 20, 17, 19], > X[8, 18, 9, 17], X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -9, 4, -2, 11, -3, 5, -7, 6, -8, 9, -4, > 8, -5, 7, -6}]
In[6]:=	Jones[L][q]
Out[6]=	3/2 5/2 7/2 9/2 11/2 13/2 -Sqrt[q] + 2 q - 6 q + 9 q - 15 q + 17 q - 18 q + 15/2 17/2 19/2 21/2 23/2 > 17 q - 13 q + 9 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 6 8 12 16 18 20 22 24 28 q + 2 q + 3 q + 6 q + 3 q + 2 q - 3 q + 2 q - 5 q - q - 30 32 34 > 2 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 21]][a, z]
Out[8]=	3 3 3 3 1 1 2 2 z z 5 z 5 z 2 z 2 z 6 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 3 z 4 z z z z > -- - ---- - ---- + -- - -- - -- 9 7 5 3 7 5 a a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 21]][a, z]
Out[9]=	-12 3 3 1 1 2 2 z 2 z 4 z 10 z 7 z a - -- + -- + ---- + ---- - ---- - ---- + --- + --- + --- + ---- + --- - 8 4 9 7 5 3 11 9 7 5 3 a a a z a z a z a z a a a a a 2 2 2 2 2 3 3 3 3 3 4 z 4 z 6 z 4 z 2 z z 7 z 12 z 15 z 20 z > ---- - ---- + ---- + ---- - ---- + --- - ---- - ----- - ----- - ----- - 12 10 8 6 4 13 11 9 7 5 a a a a a a a a a a 3 4 4 4 4 4 4 5 5 5 9 z z 9 z 4 z 19 z 19 z 6 z 4 z 16 z 22 z > ---- - --- + ---- + ---- - ----- - ----- - ---- - ---- + ----- + ----- + 3 14 12 10 8 6 4 13 11 9 a a a a a a a a a a 5 5 5 6 6 6 6 6 7 7 8 z 11 z 5 z 9 z 10 z 33 z 21 z 7 z 12 z 4 z > ---- + ----- + ---- - ---- + ----- + ----- + ----- + ---- - ----- - ---- + 7 5 3 12 10 8 6 4 11 9 a a a a a a a a a a 7 7 7 8 8 8 8 9 9 9 11 z 2 z z 10 z 13 z 5 z 2 z 5 z 7 z 2 z > ----- + ---- - -- - ----- - ----- - ---- - ---- - ---- - ---- - ---- - 7 5 3 10 8 6 4 9 7 5 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 5 q + 3 q + t + -- + -- + 6 q t + 3 q t + 9 q t + 6 q t + 8 q t + t t 12 3 12 4 14 4 14 5 16 5 16 6 > 9 q t + 10 q t + 8 q t + 7 q t + 10 q t + 6 q t + 18 6 18 7 20 7 20 8 22 8 24 9 > 7 q t + 3 q t + 6 q t + q t + 3 q t + q t

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