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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-15/2 - 4q-13/2 + 8q-11/2 - 14q-9/2 + 18q-7/2 - 22q-5/2 + 21q-3/2 - 18q-1/2 + 14q1/2 - 8q3/2 + 3q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-24 + 3q-20 - 2q-18 + 3q-16 + 5q-14 - 3q-12 + 4q-10 - q-8 + q-4 - 4q-2 + 4 - 3q2 + 4q6 - 2q8 + q12

HOMFLY-PT Polynomial:

- a-3z - a-1z-1 - a-1z + 2a-1z3 + 3az-1 + 6az + 3az3 - az5 - 4a3z-1 - 8a3z - 4a3z3 - 2a3z5 + 2a5z-1 + 3a5z + 3a5z3 - a7z

Kauffman Polynomial:

- a-3z + 2a-3z3 - a-3z5 - a-2z2 + 4a-2z4 - 3a-2z6 - a-1z-1 + 4a-1z - 7a-1z3 + 9a-1z5 - 6a-1z7 - 1 + 11z2 - 16z4 + 14z6 - 8z8 - 3az-1 + 17az - 34az3 + 26az5 - 4az7 - 5az9 - 3a2 + 24a2z2 - 53a2z4 + 51a2z6 - 19a2z8 - a2z10 - 4a3z-1 + 24a3z - 53a3z3 + 40a3z5 + a3z7 - 9a3z9 - 2a4 + 14a4z2 - 40a4z4 + 47a4z6 - 17a4z8 - a4z10 - 2a5z-1 + 15a5z - 36a5z3 + 34a5z5 - 5a5z7 - 4a5z9 - a6 + a6z2 - 5a6z4 + 12a6z6 - 6a6z8 + 3a7z - 8a7z3 + 10a7z5 - 4a7z7 - a8z2 + 2a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                       1 j = 6                     2   j = 4                   6 1   j = 2                 8 2     j = 0               10 6       j = -2             12 9         j = -4           10 9           j = -6         8 12             j = -8       6 10               j = -10     3 9                 j = -12   1 5                   j = -14   3                     j = -16 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, 82]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 82]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[16, 8, 17, 7], X[22, 13, 5, 14], > X[14, 17, 15, 18], X[20, 10, 21, 9], X[18, 21, 19, 22], X[8, 16, 9, 15], > X[10, 20, 11, 19], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -8, 6, -9, 11, -2, 4, -5, 8, -3, 5, -7, > 9, -6, 7, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 4 8 14 18 22 21 18 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 14 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 7/2 > 8 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 3 2 3 5 3 4 -8 -4 4 2 6 4 - q + --- - --- + --- + --- - --- + --- - q + q - -- - 3 q + 4 q - 20 18 16 14 12 10 2 q q q q q q q 8 12 > 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 82]][a, z]
Out[8]=	3 5 3 1 3 a 4 a 2 a z z 3 5 7 2 z -(---) + --- - ---- + ---- - -- - - + 6 a z - 8 a z + 3 a z - a z + ---- + a z z z z 3 a a a 3 3 3 5 3 5 3 5 > 3 a z - 4 a z + 3 a z - a z - 2 a z
In[9]:=	Kauffman[Link[11, Alternating, 82]][a, z]
Out[9]=	3 5 2 4 6 1 3 a 4 a 2 a z 4 z 3 -1 - 3 a - 2 a - a - --- - --- - ---- - ---- - -- + --- + 17 a z + 24 a z + a z z z z 3 a a 2 5 7 2 z 2 2 4 2 6 2 8 2 > 15 a z + 3 a z + 11 z - -- + 24 a z + 14 a z + a z - a z + 2 a 3 3 4 2 z 7 z 3 3 3 5 3 7 3 4 4 z > ---- - ---- - 34 a z - 53 a z - 36 a z - 8 a z - 16 z + ---- - 3 a 2 a a 5 5 2 4 4 4 6 4 8 4 z 9 z 5 3 5 > 53 a z - 40 a z - 5 a z + 2 a z - -- + ---- + 26 a z + 40 a z + 3 a a 6 5 5 7 5 6 3 z 2 6 4 6 6 6 > 34 a z + 10 a z + 14 z - ---- + 51 a z + 47 a z + 12 a z - 2 a 7 8 6 6 z 7 3 7 5 7 7 7 8 2 8 > a z - ---- - 4 a z + a z - 5 a z - 4 a z - 8 z - 19 a z - a 4 8 6 8 9 3 9 5 9 2 10 4 10 > 17 a z - 6 a z - 5 a z - 9 a z - 4 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	9 1 3 1 5 3 9 6 10 10 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 8 12 10 9 12 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 6 t + 8 q t + 2 q t + 6 q t + 6 3 6 2 4 2 4 2 q t q t q t q t q t 4 3 6 3 8 4 > q t + 2 q t + q t

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