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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-27/2 - 3q-25/2 + 8q-23/2 - 14q-21/2 + 20q-19/2 - 24q-17/2 + 24q-15/2 - 22q-13/2 + 16q-11/2 - 11q-9/2 + 4q-7/2 - q-5/2

A2 (sl(3)) Invariant:

- q-42 - q-40 + q-38 - 4q-36 - q-34 + 3q-32 - 4q-30 + 4q-28 + q-24 + 5q-22 - 2q-20 + 7q-18 - q-16 + 4q-12 - 3q-10 + q-8

HOMFLY-PT Polynomial:

- a5z3 - a5z5 - 2a7z-1 - 9a7z - 11a7z3 - 4a7z5 + 2a9z-1 + 4a9z - a9z3 - 2a9z5 + a11z-1 + 4a11z + 3a11z3 - a13z-1 - a13z

Kauffman Polynomial:

a5z3 - a5z5 + 3a6z4 - 4a6z6 - 2a7z-1 + 9a7z - 15a7z3 + 17a7z5 - 10a7z7 + 3a8 - 5a8z2 - 4a8z4 + 17a8z6 - 12a8z8 - 2a9z-1 + 7a9z - 15a9z3 + 17a9z5 + a9z7 - 8a9z9 + 7a10z2 - 31a10z4 + 45a10z6 - 19a10z8 - 2a10z10 + a11z-1 - 4a11z + 2a11z3 - 3a11z5 + 18a11z7 - 13a11z9 - 3a12 + 15a12z2 - 31a12z4 + 34a12z6 - 12a12z8 - 2a12z10 + a13z-1 - a13z - 4a13z3 + 5a13z5 + 4a13z7 - 5a13z9 - 4a14z4 + 9a14z6 - 5a14z8 + a15z - 5a15z3 + 7a15z5 - 3a15z7 + a16 - 3a16z2 + 3a16z4 - 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                     4 1 j = -8                   7     j = -10                 9 4     j = -12               13 7       j = -14             12 10         j = -16           12 12           j = -18         8 12             j = -20       6 12               j = -22     2 8                 j = -24   1 6                   j = -26   2                     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, 81]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 81]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[18, 13, 19, 14], X[10, 17, 11, 18], > X[8, 21, 9, 22], X[16, 7, 17, 8], X[20, 9, 21, 10], X[22, 15, 5, 16], > X[14, 19, 15, 20], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 6, -5, 7, -4, 11, -2, 3, -9, 8, -6, 4, -3, > 9, -7, 5, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(27/2) 3 8 14 20 24 24 22 16 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 11 4 -(5/2) > ---- + ---- - q 9/2 7/2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 -40 -38 4 -34 3 4 4 -24 5 2 7 -q - q + q - --- - q + --- - --- + --- + q + --- - --- + --- - 36 32 30 28 22 20 18 q q q q q q q -16 4 3 -8 > q + --- - --- + q 12 10 q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 81]][a, z]
Out[8]=	7 9 11 13 -2 a 2 a a a 7 9 11 13 5 3 ----- + ---- + --- - --- - 9 a z + 4 a z + 4 a z - a z - a z - z z z z 7 3 9 3 11 3 5 5 7 5 9 5 > 11 a z - a z + 3 a z - a z - 4 a z - 2 a z
In[9]:=	Kauffman[Link[11, Alternating, 81]][a, z]
Out[9]=	7 9 11 13 8 12 16 2 a 2 a a a 7 9 11 3 a - 3 a + a - ---- - ---- + --- + --- + 9 a z + 7 a z - 4 a z - z z z z 13 15 8 2 10 2 12 2 16 2 5 3 > a z + a z - 5 a z + 7 a z + 15 a z - 3 a z + a z - 7 3 9 3 11 3 13 3 15 3 6 4 8 4 > 15 a z - 15 a z + 2 a z - 4 a z - 5 a z + 3 a z - 4 a z - 10 4 12 4 14 4 16 4 5 5 7 5 9 5 > 31 a z - 31 a z - 4 a z + 3 a z - a z + 17 a z + 17 a z - 11 5 13 5 15 5 6 6 8 6 10 6 > 3 a z + 5 a z + 7 a z - 4 a z + 17 a z + 45 a z + 12 6 14 6 16 6 7 7 9 7 11 7 13 7 > 34 a z + 9 a z - a z - 10 a z + a z + 18 a z + 4 a z - 15 7 8 8 10 8 12 8 14 8 9 9 > 3 a z - 12 a z - 19 a z - 12 a z - 5 a z - 8 a z - 11 9 13 9 10 10 12 10 > 13 a z - 5 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 2 1 6 2 8 6 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 12 8 12 12 12 12 10 13 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 7 9 4 7 4 > ------ + ------ + ------ + ----- + ---- 12 3 10 3 10 2 8 2 6 q t q t q t q t q t

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