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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-21/2 - 3q-19/2 + 7q-17/2 - 13q-15/2 + 17q-13/2 - 21q-11/2 + 21q-9/2 - 19q-7/2 + 14q-5/2 - 9q-3/2 + 4q-1/2 - q1/2

A2 (sl(3)) Invariant:

- q-34 - 2q-32 + q-30 - 2q-26 + 6q-24 + q-22 + q-20 + 5q-18 - 2q-16 + 3q-14 - 3q-12 + 3q-8 - 4q-6 + 4q-4 - 2 + q2

HOMFLY-PT Polynomial:

- az3 - 2a3z - a3z3 + a3z5 + a5z + 2a5z3 + 2a5z5 - 2a7z-1 - 6a7z - 5a7z3 + 3a9z-1 + 4a9z - a11z-1

Kauffman Polynomial:

az3 - az5 + 5a2z4 - 4a2z6 + 2a3z - 6a3z3 + 13a3z5 - 8a3z7 + 5a4z2 - 15a4z4 + 19a4z6 - 10a4z8 - 5a5z3 + 2a5z5 + 7a5z7 - 7a5z9 + 8a6z2 - 31a6z4 + 34a6z6 - 12a6z8 - 2a6z10 - 2a7z-1 + 7a7z - 6a7z3 - 9a7z5 + 20a7z7 - 11a7z9 + 3a8 - 3a8z2 - 8a8z4 + 17a8z6 - 6a8z8 - 2a8z10 - 3a9z-1 + 12a9z - 15a9z3 + 11a9z5 + 2a9z7 - 4a9z9 + 3a10 - 9a10z2 + 6a10z4 + 5a10z6 - 4a10z8 - a11z-1 + 3a11z - 7a11z3 + 8a11z5 - 3a11z7 + a12 - 3a12z2 + 3a12z4 - a12z6

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 2                       1 j = 0                     3   j = -2                   6 1   j = -4                 9 4     j = -6               10 5       j = -8             11 9         j = -10           10 10           j = -12         7 11             j = -14       6 10               j = -16     2 8                 j = -18   1 5                   j = -20   2                     j = -22 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, 231]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 231]]
Out[4]=	PD[X[8, 1, 9, 2], X[12, 3, 13, 4], X[22, 13, 7, 14], X[14, 9, 15, 10], > X[10, 21, 11, 22], X[18, 5, 19, 6], X[20, 16, 21, 15], X[16, 20, 17, 19], > X[2, 7, 3, 8], X[4, 11, 5, 12], X[6, 17, 1, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10, 6, -11}, > {9, -1, 4, -5, 10, -2, 3, -4, 7, -8, 11, -6, 8, -7, 5, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 3 7 13 17 21 21 19 14 9 q - ----- + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q q 4 > ------- - Sqrt[q] Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 2 -30 2 6 -22 -20 5 2 3 3 3 -2 - q - --- + q - --- + --- + q + q + --- - --- + --- - --- + -- - 32 26 24 18 16 14 12 8 q q q q q q q q 4 4 2 > -- + -- + q 6 4 q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 231]][a, z]
Out[8]=	7 9 11 -2 a 3 a a 3 5 7 9 3 3 3 5 3 ----- + ---- - --- - 2 a z + a z - 6 a z + 4 a z - a z - a z + 2 a z - z z z 7 3 3 5 5 5 > 5 a z + a z + 2 a z
In[9]:=	Kauffman[Link[11, Alternating, 231]][a, z]
Out[9]=	7 9 11 8 10 12 2 a 3 a a 3 7 9 11 3 a + 3 a + a - ---- - ---- - --- + 2 a z + 7 a z + 12 a z + 3 a z + z z z 4 2 6 2 8 2 10 2 12 2 3 3 3 > 5 a z + 8 a z - 3 a z - 9 a z - 3 a z + a z - 6 a z - 5 3 7 3 9 3 11 3 2 4 4 4 6 4 > 5 a z - 6 a z - 15 a z - 7 a z + 5 a z - 15 a z - 31 a z - 8 4 10 4 12 4 5 3 5 5 5 7 5 > 8 a z + 6 a z + 3 a z - a z + 13 a z + 2 a z - 9 a z + 9 5 11 5 2 6 4 6 6 6 8 6 10 6 > 11 a z + 8 a z - 4 a z + 19 a z + 34 a z + 17 a z + 5 a z - 12 6 3 7 5 7 7 7 9 7 11 7 4 8 > a z - 8 a z + 7 a z + 20 a z + 2 a z - 3 a z - 10 a z - 6 8 8 8 10 8 5 9 7 9 9 9 6 10 > 12 a z - 6 a z - 4 a z - 7 a z - 11 a z - 4 a z - 2 a z - 8 10 > 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 6 1 2 1 5 2 8 6 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 22 9 20 8 18 8 18 7 16 7 16 6 14 6 q q q t q t q t q t q t q t q t 10 7 11 10 10 11 9 10 5 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 14 5 12 5 12 4 10 4 10 3 8 3 8 2 6 2 6 q t q t q t q t q t q t q t q t q t 9 t 2 2 > ---- + 3 t + -- + q t 4 2 q t q

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