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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X7,16,8,17 X9,18,10,19 X17,8,18,9 X19,22,20,5 X13,20,14,21 X21,14,22,15 X15,10,16,11 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

q-27/2 - 2q-25/2 + 3q-23/2 - 3q-21/2 + 4q-19/2 - 4q-17/2 + 2q-15/2 - 2q-13/2 - q-7/2

A2 (sl(3)) Invariant:

- q-42 - q-40 - 2q-36 - q-32 - q-30 + q-28 + q-26 + 4q-24 + 2q-22 + 2q-20 + 2q-18 + q-16 + q-14 + q-12

HOMFLY-PT Polynomial:

- 2a7z-1 - 9a7z - 14a7z3 - 7a7z5 - a7z7 + 2a9z-1 + 4a9z + 2a9z3 + a11z-1 + 4a11z + 2a11z3 - a13z-1 - a13z

Kauffman Polynomial:

- 2a7z-1 + 9a7z - 14a7z3 + 7a7z5 - a7z7 + 3a8 - 5a8z2 + 2a8z4 - 2a9z-1 + 7a9z - 7a9z3 + 2a9z5 + 7a10z2 - 14a10z4 + 7a10z6 - a10z8 + a11z-1 - 4a11z + 10a11z3 - 12a11z5 + 6a11z7 - a11z9 - 3a12 + 15a12z2 - 26a12z4 + 16a12z6 - 3a12z8 + a13z-1 - a13z - 3a13z3 + a13z5 + 3a13z7 - a13z9 - 6a14z4 + 8a14z6 - 2a14z8 + a15z - 6a15z3 + 8a15z5 - 2a15z7 + a16 - 3a16z2 + 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 = -6                       1 j = -8                       1 j = -10                 1 1     j = -12               2         j = -14             2 3 1       j = -16           3 1           j = -18         2 3 1           j = -20       2 3               j = -22     1 2 1               j = -24   1 2                   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, NonAlternating, 95]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 95]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[7, 16, 8, 17], X[9, 18, 10, 19], > X[17, 8, 18, 9], X[19, 22, 20, 5], X[13, 20, 14, 21], X[21, 14, 22, 15], > X[15, 10, 16, 11], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -3, 5, -4, 9, 11, -2, -7, 8, -9, 3, -5, 4, > -6, 7, -8, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-(27/2) 2 3 3 4 4 2 2 -(7/2) q - ----- + ----- - ----- + ----- - ----- + ----- - ----- - q 25/2 23/2 21/2 19/2 17/2 15/2 13/2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 -40 2 -32 -30 -28 -26 4 2 2 2 -16 -q - q - --- - q - q + q + q + --- + --- + --- + --- + q + 36 24 22 20 18 q q q q q -14 -12 > q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 95]][a, z]
Out[8]=	7 9 11 13 -2 a 2 a a a 7 9 11 13 7 3 ----- + ---- + --- - --- - 9 a z + 4 a z + 4 a z - a z - 14 a z + z z z z 9 3 11 3 7 5 7 7 > 2 a z + 2 a z - 7 a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 95]][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 7 3 > a z + a z - 5 a z + 7 a z + 15 a z - 3 a z - 14 a z - 9 3 11 3 13 3 15 3 8 4 10 4 > 7 a z + 10 a z - 3 a z - 6 a z + 2 a z - 14 a z - 12 4 14 4 16 4 7 5 9 5 11 5 13 5 > 26 a z - 6 a z + 4 a z + 7 a z + 2 a z - 12 a z + a z + 15 5 10 6 12 6 14 6 16 6 7 7 11 7 > 8 a z + 7 a z + 16 a z + 8 a z - a z - a z + 6 a z + 13 7 15 7 10 8 12 8 14 8 11 9 13 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 1 1 1 2 1 2 2 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 1 3 2 3 3 1 1 2 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 22 7 20 7 18 7 18 6 16 6 18 5 16 5 14 5 q t q t q t q t q t q t q t q t 3 2 1 1 1 > ------ + ------ + ------ + ------ + ------ 14 4 12 4 14 3 10 3 10 2 q t q t q t q t q t

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