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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-1/2 + 2q1/2 - 5q3/2 + 7q5/2 - 11q7/2 + 12q9/2 - 13q11/2 + 11q13/2 - 8q15/2 + 6q17/2 - 3q19/2 + q21/2

A2 (sl(3)) Invariant:

q-2 + q2 + 3q4 - q6 + 3q8 + 2q10 + 3q14 + 2q18 - 2q22 + q24 - 3q26 - q28 + q30 - q32

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- a-12z2 + 3a-12z4 - a-12z6 - 4a-11z3 + 9a-11z5 - 3a-11z7 - 3a-10 + 10a-10z2 - 19a-10z4 + 18a-10z6 - 5a-10z8 + a-9z-1 - 3a-9z + 8a-9z3 - 14a-9z5 + 13a-9z7 - 4a-9z9 - 7a-8 + 27a-8z2 - 44a-8z4 + 28a-8z6 - 5a-8z8 - a-8z10 + 2a-7z-1 - 8a-7z + 20a-7z3 - 32a-7z5 + 21a-7z7 - 6a-7z9 - 4a-6 + 17a-6z2 - 23a-6z4 + 11a-6z6 - 2a-6z8 - a-6z10 + a-5z-1 - 7a-5z + 12a-5z3 - 8a-5z5 + 3a-5z7 - 2a-5z9 + 3a-4z4 - 2a-4z8 + a-3z-1 - 5a-3z + 7a-3z3 - 2a-3z7 - a-2 - a-2z2 + 4a-2z4 - 2a-2z6 + a-1z-1 - 3a-1z + 3a-1z3 - a-1z5

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 = 22                       1 j = 20                     2   j = 18                   4 1   j = 16                 4 2     j = 14               7 4       j = 12             6 4         j = 10           6 7           j = 8         5 6             j = 6       2 6               j = 4     3 5                 j = 2   1 4                   j = 0   1                     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, 22]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 22]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[12, 8, 13, 7], X[18, 10, 19, 9], > X[22, 14, 5, 13], X[20, 16, 21, 15], X[16, 20, 17, 19], X[14, 22, 15, 21], > 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, -8, 6, -7, 9, -4, > 7, -6, 8, -5}]
In[6]:=	Jones[L][q]
Out[6]=	1 3/2 5/2 7/2 9/2 11/2 -(-------) + 2 Sqrt[q] - 5 q + 7 q - 11 q + 12 q - 13 q + Sqrt[q] 13/2 15/2 17/2 19/2 21/2 > 11 q - 8 q + 6 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-2 2 4 6 8 10 14 18 22 24 26 q + q + 3 q - q + 3 q + 2 q + 3 q + 2 q - 2 q + q - 3 q - 28 30 32 > q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 22]][a, z]
Out[8]=	3 3 1 2 1 1 1 z 3 z 2 z 2 z 2 z z 2 z ---- - ---- + ---- - ---- + --- + -- - --- + --- - --- + --- + -- - ---- - 9 7 5 3 a z 9 7 5 3 a 9 7 a z a z a z a z a a a a a a 3 3 5 5 5 2 z z z z z > ---- + -- - -- - -- - -- 3 a 7 5 3 a a a a
In[9]:=	Kauffman[Link[11, Alternating, 22]][a, z]
Out[9]=	-3 7 4 -2 1 2 1 1 1 3 z 8 z 7 z 5 z --- - -- - -- - a + ---- + ---- + ---- + ---- + --- - --- - --- - --- - --- - 10 8 6 9 7 5 3 a z 9 7 5 3 a a a a z a z a z a z a a a a 2 2 2 2 2 3 3 3 3 3 z z 10 z 27 z 17 z z 4 z 8 z 20 z 12 z > --- - --- + ----- + ----- + ----- - -- - ---- + ---- + ----- + ----- + a 12 10 8 6 2 11 9 7 5 a a a a a a a a a 3 3 4 4 4 4 4 4 5 5 7 z 3 z 3 z 19 z 44 z 23 z 3 z 4 z 9 z 14 z > ---- + ---- + ---- - ----- - ----- - ----- + ---- + ---- + ---- - ----- - 3 a 12 10 8 6 4 2 11 9 a a a a a a a a a 5 5 5 6 6 6 6 6 7 7 32 z 8 z z z 18 z 28 z 11 z 2 z 3 z 13 z > ----- - ---- - -- - --- + ----- + ----- + ----- - ---- - ---- + ----- + 7 5 a 12 10 8 6 2 11 9 a a a a a a a a a 7 7 7 8 8 8 8 9 9 9 21 z 3 z 2 z 5 z 5 z 2 z 2 z 4 z 6 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 2 4 1 1 q 4 6 6 2 8 2 8 3 4 q + 3 q + ----- + - + -- + 5 q t + 2 q t + 6 q t + 5 q t + 6 q t + 2 2 t t q t 10 3 10 4 12 4 12 5 14 5 14 6 > 6 q t + 7 q t + 6 q t + 4 q t + 7 q t + 4 q t + 16 6 16 7 18 7 18 8 20 8 22 9 > 4 q t + 2 q t + 4 q t + q t + 2 q t + q t

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