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DrawMorseLink

PD Presentation:

X6172 X18,7,19,8 X4,19,1,20 X5,14,6,15 X3849 X9,16,10,17 X15,10,16,11 X11,20,12,21 X13,22,14,5 X21,12,22,13 X17,2,18,3

Gauss Code:

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

Jones Polynomial:

2q-25/2 - 3q-23/2 + 5q-21/2 - 6q-19/2 + 5q-17/2 - 6q-15/2 + 4q-13/2 - 3q-11/2 + q-9/2 - q-7/2

A2 (sl(3)) Invariant:

- q-42 - q-40 - 3q-38 - 2q-36 - q-34 + 3q-30 + q-28 + 4q-26 + 2q-24 + 2q-22 + 2q-20 + 2q-16 + q-12

HOMFLY-PT Polynomial:

- a7z-1 - 6a7z - 11a7z3 - 6a7z5 - a7z7 - a9z-1 - 3a9z - 7a9z3 - 5a9z5 - a9z7 + 4a11z-1 + 8a11z + 5a11z3 + a11z5 - 2a13z-1 - a13z

Kauffman Polynomial:

- a7z-1 + 6a7z - 11a7z3 + 6a7z5 - a7z7 + a8 - 2a8z2 - 3a8z4 + 4a8z6 - a8z8 + a9z-1 - 3a9z + 4a9z3 - 5a9z5 + 4a9z7 - a9z9 - 5a10 + 15a10z2 - 17a10z4 + 12a10z6 - 3a10z8 + 4a11z-1 - 16a11z + 22a11z3 - 9a11z5 + 3a11z7 - a11z9 - 6a12 + 17a12z2 - 12a12z4 + 6a12z6 - 2a12z8 + 2a13z-1 - 8a13z + 6a13z3 + a13z5 - 2a13z7 + a14 - 3a14z2 + 2a14z4 - 2a14z6 - a15z - a15z3 - a15z5 + 2a16 - 3a16z2

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -6                   1 j = -8                 1 1 j = -10               2     j = -12             2 1     j = -14           4 2       j = -16         2 3         j = -18       4 3           j = -20     1 2             j = -22   2 4               j = -24   1                 j = -26 2

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, 47]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 47]]
Out[4]=	PD[X[6, 1, 7, 2], X[18, 7, 19, 8], X[4, 19, 1, 20], X[5, 14, 6, 15], > X[3, 8, 4, 9], X[9, 16, 10, 17], X[15, 10, 16, 11], X[11, 20, 12, 21], > X[13, 22, 14, 5], X[21, 12, 22, 13], X[17, 2, 18, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 11, -5, -3}, {-4, -1, 2, 5, -6, 7, -8, 10, -9, 4, -7, 6, -11, -2, > 3, 8, -10, 9}]
In[6]:=	Jones[L][q]
Out[6]=	2 3 5 6 5 6 4 3 -(9/2) ----- - ----- + ----- - ----- + ----- - ----- + ----- - ----- + q - 25/2 23/2 21/2 19/2 17/2 15/2 13/2 11/2 q q q q q q q q -(7/2) > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 -40 3 2 -34 3 -28 4 2 2 2 2 -q - q - --- - --- - q + --- + q + --- + --- + --- + --- + --- + 38 36 30 26 24 22 20 16 q q q q q q q q -12 > q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 47]][a, z]
Out[8]=	7 9 11 13 a a 4 a 2 a 7 9 11 13 7 3 -(--) - -- + ----- - ----- - 6 a z - 3 a z + 8 a z - a z - 11 a z - z z z z 9 3 11 3 7 5 9 5 11 5 7 7 9 7 > 7 a z + 5 a z - 6 a z - 5 a z + a z - a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 47]][a, z]
Out[9]=	7 9 11 13 8 10 12 14 16 a a 4 a 2 a 7 9 a - 5 a - 6 a + a + 2 a - -- + -- + ----- + ----- + 6 a z - 3 a z - z z z z 11 13 15 8 2 10 2 12 2 14 2 > 16 a z - 8 a z - a z - 2 a z + 15 a z + 17 a z - 3 a z - 16 2 7 3 9 3 11 3 13 3 15 3 8 4 > 3 a z - 11 a z + 4 a z + 22 a z + 6 a z - a z - 3 a z - 10 4 12 4 14 4 7 5 9 5 11 5 13 5 > 17 a z - 12 a z + 2 a z + 6 a z - 5 a z - 9 a z + a z - 15 5 8 6 10 6 12 6 14 6 7 7 9 7 > a z + 4 a z + 12 a z + 6 a z - 2 a z - a z + 4 a z + 11 7 13 7 8 8 10 8 12 8 9 9 11 9 > 3 a z - 2 a z - a z - 3 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-8 -6 2 1 2 4 1 2 4 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 26 9 24 8 22 8 22 7 20 7 20 6 18 6 q t q t q t q t q t q t q t 3 2 3 4 2 2 1 2 1 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + ---- 18 5 16 5 16 4 14 4 14 3 12 3 12 2 10 2 8 q t q t q t q t q t q t q t q t q t

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