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DrawMorseLink

PD Presentation:

X6172 X3,12,4,13 X7,18,8,19 X19,22,20,5 X9,21,10,20 X21,9,22,8 X16,11,17,12 X14,17,15,18 X10,15,11,16 X2536 X13,4,14,1

Gauss Code:

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

Jones Polynomial:

q-21/2 - 2q-19/2 + 2q-17/2 - 2q-15/2 + q-13/2 - q-11/2 + q-7/2 - 2q-5/2 + q-3/2 - q-1/2

A2 (sl(3)) Invariant:

- q-34 - 2q-32 + q-30 + 2q-26 + 3q-24 + q-22 + q-20 - 2q-18 - q-14 + 2q-10 + 2q-8 + q-6 + q-4 + q-2

HOMFLY-PT Polynomial:

- 2a3z-1 - 6a3z - 5a3z3 - a3z5 + 5a5z-1 + 11a5z + 11a5z3 + 6a5z5 + a5z7 - 6a7z-1 - 11a7z - 7a7z3 - a7z5 + 4a9z-1 + 4a9z - a11z-1

Kauffman Polynomial:

- 2a3z-1 + 8a3z - 11a3z3 + 6a3z5 - a3z7 + a4 - 5a4z4 + 5a4z6 - a4z8 - 5a5z-1 + 22a5z - 33a5z3 + 19a5z5 - 3a5z7 + a6 - a6z2 - 6a6z4 + 6a6z6 - a6z8 - 6a7z-1 + 28a7z - 42a7z3 + 22a7z5 - 3a7z7 + 3a8 - 7a8z2 - a8z4 + 5a8z6 - a8z8 - 4a9z-1 + 18a9z - 29a9z3 + 18a9z5 - 3a9z7 + 3a10 - 9a10z2 + 4a10z4 + 3a10z6 - a10z8 - a11z-1 + 4a11z - 9a11z3 + 9a11z5 - 2a11z7 + a12 - 3a12z2 + 4a12z4 - 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 = 0                       1 j = -2                         j = -4                   2 1   j = -6               1 1 1     j = -8             1 1 1       j = -10           2 2 1         j = -12         2 3 1           j = -14       1 1 1             j = -16     1 2 1               j = -18   1 1                   j = -20   1                     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, NonAlternating, 111]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 111]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 12, 4, 13], X[7, 18, 8, 19], X[19, 22, 20, 5], > X[9, 21, 10, 20], X[21, 9, 22, 8], X[16, 11, 17, 12], X[14, 17, 15, 18], > X[10, 15, 11, 16], X[2, 5, 3, 6], X[13, 4, 14, 1]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -2, 11}, {10, -1, -3, 6, -5, -9, 7, 2, -11, -8, 9, -7, 8, 3, > -4, 5, -6, 4}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 2 2 2 -(13/2) -(11/2) -(7/2) 2 q - ----- + ----- - ----- + q - q + q - ---- + 19/2 17/2 15/2 5/2 q q q q -(3/2) 1 > q - ------- Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 2 -30 2 3 -22 -20 2 -14 2 2 -6 -q - --- + q + --- + --- + q + q - --- - q + --- + -- + q + 32 26 24 18 10 8 q q q q q q -4 -2 > q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 111]][a, z]
Out[8]=	3 5 7 9 11 -2 a 5 a 6 a 4 a a 3 5 7 9 ----- + ---- - ---- + ---- - --- - 6 a z + 11 a z - 11 a z + 4 a z - z z z z z 3 3 5 3 7 3 3 5 5 5 7 5 5 7 > 5 a z + 11 a z - 7 a z - a z + 6 a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 111]][a, z]
Out[9]=	3 5 7 9 11 4 6 8 10 12 2 a 5 a 6 a 4 a a 3 a + a + 3 a + 3 a + a - ---- - ---- - ---- - ---- - --- + 8 a z + z z z z z 5 7 9 11 6 2 8 2 10 2 > 22 a z + 28 a z + 18 a z + 4 a z - a z - 7 a z - 9 a z - 12 2 3 3 5 3 7 3 9 3 11 3 4 4 > 3 a z - 11 a z - 33 a z - 42 a z - 29 a z - 9 a z - 5 a z - 6 4 8 4 10 4 12 4 3 5 5 5 7 5 > 6 a z - a z + 4 a z + 4 a z + 6 a z + 19 a z + 22 a z + 9 5 11 5 4 6 6 6 8 6 10 6 12 6 > 18 a z + 9 a z + 5 a z + 6 a z + 5 a z + 3 a z - a z - 3 7 5 7 7 7 9 7 11 7 4 8 6 8 8 8 > a z - 3 a z - 3 a z - 3 a z - 2 a z - a z - a z - a z - 10 8 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 2 1 1 1 1 1 2 1 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 22 9 20 8 18 8 18 7 16 7 16 6 14 6 q q t q t q t q t q t q t q t 1 1 2 1 3 2 1 2 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 16 5 14 5 12 5 14 4 12 4 10 4 12 3 10 3 q t q t q t q t q t q t q t q t 1 1 1 1 1 1 t 2 > ----- + ------ + ----- + ----- + ---- + ---- + -- + t 8 3 10 2 8 2 6 2 8 6 4 q t q t q t q t q t q t q

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