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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-27/2 - 2q-25/2 + 5q-23/2 - 7q-21/2 + 9q-19/2 - 11q-17/2 + 10q-15/2 - 10q-13/2 + 6q-11/2 - 4q-9/2 + 2q-7/2 - q-5/2

A2 (sl(3)) Invariant:

- q-42 - 2q-40 - q-38 - 3q-36 - q-34 + q-32 + 4q-28 + 2q-26 + 3q-24 + 3q-22 + 3q-18 + q-12 - q-10 + q-8

HOMFLY-PT Polynomial:

- a5z - 3a5z3 - a5z5 - a7z-1 - 4a7z - 6a7z3 - 2a7z5 - a9z-1 - 4a9z - 6a9z3 - 2a9z5 + 4a11z-1 + 8a11z + 3a11z3 - 2a13z-1 - a13z

Kauffman Polynomial:

- a5z + 3a5z3 - a5z5 - a6z2 + 5a6z4 - 2a6z6 - a7z-1 + 5a7z - 9a7z3 + 9a7z5 - 3a7z7 + a8 - 2a8z2 - 6a8z4 + 8a8z6 - 3a8z8 + a9z-1 - 2a9z + 6a9z3 - 17a9z5 + 11a9z7 - 3a9z9 - 5a10 + 18a10z2 - 29a10z4 + 13a10z6 - a10z8 - a10z10 + 4a11z-1 - 15a11z + 34a11z3 - 40a11z5 + 21a11z7 - 5a11z9 - 6a12 + 19a12z2 - 17a12z4 + 7a12z6 - a12z10 + 2a13z-1 - 8a13z + 13a13z3 - 7a13z5 + 5a13z7 - 2a13z9 + a14 - 5a14z2 + 5a14z4 + 3a14z6 - 2a14z8 - a15z - 3a15z3 + 6a15z5 - 2a15z7 + 2a16 - 5a16z2 + 4a16z4 - a16z6

Khovanov Homology:

trqj r = -11 r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -4                       1 j = -6                     2 1 j = -8                   2     j = -10                 4 2     j = -12               6 2       j = -14             5 5         j = -16           6 5           j = -18         3 5             j = -20       4 6               j = -22     1 3                 j = -24   1 4                   j = -26   1                     j = -28 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, 75]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 75]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[20, 15, 21, 16], X[16, 7, 17, 8], > X[18, 9, 19, 10], X[8, 17, 9, 18], X[10, 19, 11, 20], X[22, 13, 5, 14], > X[14, 21, 15, 22], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, -6, 5, -7, 11, -2, 8, -9, 3, -4, 6, -5, > 7, -3, 9, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(27/2) 2 5 7 9 11 10 10 6 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 4 2 -(5/2) > ---- + ---- - q 9/2 7/2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 2 -38 3 -34 -32 4 2 3 3 3 -12 -q - --- - q - --- - q + q + --- + --- + --- + --- + --- + q - 40 36 28 26 24 22 18 q q q q q q q -10 -8 > q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 75]][a, z]
Out[8]=	7 9 11 13 a a 4 a 2 a 5 7 9 11 13 -(--) - -- + ----- - ----- - a z - 4 a z - 4 a z + 8 a z - a z - z z z z 5 3 7 3 9 3 11 3 5 5 7 5 9 5 > 3 a z - 6 a z - 6 a z + 3 a z - a z - 2 a z - 2 a z
In[9]:=	Kauffman[Link[11, Alternating, 75]][a, z]
Out[9]=	7 9 11 13 8 10 12 14 16 a a 4 a 2 a 5 7 a - 5 a - 6 a + a + 2 a - -- + -- + ----- + ----- - a z + 5 a z - z z z z 9 11 13 15 6 2 8 2 10 2 > 2 a z - 15 a z - 8 a z - a z - a z - 2 a z + 18 a z + 12 2 14 2 16 2 5 3 7 3 9 3 11 3 > 19 a z - 5 a z - 5 a z + 3 a z - 9 a z + 6 a z + 34 a z + 13 3 15 3 6 4 8 4 10 4 12 4 > 13 a z - 3 a z + 5 a z - 6 a z - 29 a z - 17 a z + 14 4 16 4 5 5 7 5 9 5 11 5 13 5 > 5 a z + 4 a z - a z + 9 a z - 17 a z - 40 a z - 7 a z + 15 5 6 6 8 6 10 6 12 6 14 6 16 6 > 6 a z - 2 a z + 8 a z + 13 a z + 7 a z + 3 a z - a z - 7 7 9 7 11 7 13 7 15 7 8 8 10 8 > 3 a z + 11 a z + 21 a z + 5 a z - 2 a z - 3 a z - a z - 14 8 9 9 11 9 13 9 10 10 12 10 > 2 a z - 3 a z - 5 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 1 1 4 1 3 4 q + q + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 28 11 26 10 24 10 24 9 22 9 22 8 20 8 q t q t q t q t q t q t q t 6 3 5 6 5 5 5 6 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 20 7 18 7 18 6 16 6 16 5 14 5 14 4 12 4 q t q t q t q t q t q t q t q t 2 4 2 2 2 > ------ + ------ + ------ + ----- + ---- 12 3 10 3 10 2 8 2 6 q t q t q t q t q t

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