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DrawMorseLink

PD Presentation:

X6172 X20,7,21,8 X4,21,1,22 X14,12,15,11 X8493 X12,5,13,6 X22,13,5,14 X18,15,19,16 X16,9,17,10 X10,17,11,18 X2,20,3,19

Gauss Code:

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

Jones Polynomial:

- q-19/2 + 4q-17/2 - 8q-15/2 + 15q-13/2 - 21q-11/2 + 24q-9/2 - 25q-7/2 + 21q-5/2 - 17q-3/2 + 10q-1/2 - 5q1/2 + q3/2

A2 (sl(3)) Invariant:

q-30 - 3q-26 + q-24 - q-22 - 4q-20 + 4q-18 - 3q-16 + 2q-14 + 2q-12 - q-10 + 7q-8 - 3q-6 + 5q-4 + 2q-2 - 2 + 3q2 - q4

HOMFLY-PT Polynomial:

- az-1 - az + az3 + az5 - 2a3z - 4a3z3 - 3a3z5 - a3z7 + 2a5z-1 + 6a5z + 7a5z3 + 3a5z5 - a7z-1 - 4a7z - 3a7z3 + a9z

Kauffman Polynomial:

z4 - z6 + az-1 - az - 5az3 + 10az5 - 5az7 - a2 + 2a2z2 - 10a2z4 + 19a2z6 - 9a2z8 + 4a3z - 18a3z3 + 21a3z5 + 3a3z7 - 7a3z9 + 3a4 - 27a4z4 + 47a4z6 - 18a4z8 - 2a4z10 - 2a5z-1 + 11a5z - 31a5z3 + 26a5z5 + 10a5z7 - 13a5z9 + 5a6 - 8a6z2 - 15a6z4 + 36a6z6 - 17a6z8 - 2a6z10 - a7z-1 + 8a7z - 24a7z3 + 25a7z5 - 5a7z7 - 6a7z9 + 2a8 - 9a8z2 + 7a8z4 + 5a8z6 - 8a8z8 + 2a9z - 5a9z3 + 9a9z5 - 7a9z7 - 3a10z2 + 6a10z4 - 4a10z6 + a11z3 - a11z5

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 4                       1 j = 2                     4   j = 0                   6 1   j = -2                 11 4     j = -4               12 8       j = -6             13 9         j = -8           11 12           j = -10         10 13             j = -12       5 11               j = -14     3 10                 j = -16   1 5                   j = -18   3                     j = -20 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, 12]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 12]]
Out[4]=	PD[X[6, 1, 7, 2], X[20, 7, 21, 8], X[4, 21, 1, 22], X[14, 12, 15, 11], > X[8, 4, 9, 3], X[12, 5, 13, 6], X[22, 13, 5, 14], X[18, 15, 19, 16], > X[16, 9, 17, 10], X[10, 17, 11, 18], X[2, 20, 3, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {6, -1, 2, -5, 9, -10, 4, -6, 7, -4, 8, -9, 10, -8, > 11, -2, 3, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 4 8 15 21 24 25 21 17 -q + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q 10 3/2 > ------- - 5 Sqrt[q] + q Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 3 -24 -22 4 4 3 2 2 -10 7 3 -2 + q - --- + q - q - --- + --- - --- + --- + --- - q + -- - -- + 26 20 18 16 14 12 8 6 q q q q q q q q 5 2 2 4 > -- + -- + 3 q - q 4 2 q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 12]][a, z]
Out[8]=	5 7 a 2 a a 3 5 7 9 3 3 3 -(-) + ---- - -- - a z - 2 a z + 6 a z - 4 a z + a z + a z - 4 a z + z z z 5 3 7 3 5 3 5 5 5 3 7 > 7 a z - 3 a z + a z - 3 a z + 3 a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 12]][a, z]
Out[9]=	5 7 2 4 6 8 a 2 a a 3 5 7 -a + 3 a + 5 a + 2 a + - - ---- - -- - a z + 4 a z + 11 a z + 8 a z + z z z 9 2 2 6 2 8 2 10 2 3 3 3 > 2 a z + 2 a z - 8 a z - 9 a z - 3 a z - 5 a z - 18 a z - 5 3 7 3 9 3 11 3 4 2 4 4 4 > 31 a z - 24 a z - 5 a z + a z + z - 10 a z - 27 a z - 6 4 8 4 10 4 5 3 5 5 5 7 5 > 15 a z + 7 a z + 6 a z + 10 a z + 21 a z + 26 a z + 25 a z + 9 5 11 5 6 2 6 4 6 6 6 8 6 > 9 a z - a z - z + 19 a z + 47 a z + 36 a z + 5 a z - 10 6 7 3 7 5 7 7 7 9 7 2 8 > 4 a z - 5 a z + 3 a z + 10 a z - 5 a z - 7 a z - 9 a z - 4 8 6 8 8 8 3 9 5 9 7 9 4 10 > 18 a z - 17 a z - 8 a z - 7 a z - 13 a z - 6 a z - 2 a z - 6 10 > 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	8 11 1 3 1 5 3 10 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 20 8 18 7 16 7 16 6 14 6 14 5 12 5 q q q t q t q t q t q t q t q t 11 10 13 11 12 13 9 12 > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + 6 t + 12 4 10 4 10 3 8 3 8 2 6 2 6 4 q t q t q t q t q t q t q t q t 4 t 2 2 2 4 3 > --- + t + 4 q t + q t 2 q

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