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DrawMorseLink

PD Presentation:

X8192 X10,3,11,4 X14,17,15,18 X16,5,17,6 X4,15,5,16 X18,11,19,12 X22,19,7,20 X20,14,21,13 X12,22,13,21 X2738 X6,9,1,10

Gauss Code:

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

Jones Polynomial:

q-21/2 - 4q-19/2 + 7q-17/2 - 12q-15/2 + 16q-13/2 - 18q-11/2 + 18q-9/2 - 16q-7/2 + 11q-5/2 - 7q-3/2 + 3q-1/2 - q1/2

A2 (sl(3)) Invariant:

- q-32 + 2q-30 + q-28 - q-26 + 5q-24 - q-22 - q-20 + q-18 - 3q-16 + 3q-14 - q-12 + 2q-10 + 3q-8 - 3q-6 + 3q-4 - 1 + q2

HOMFLY-PT Polynomial:

- az - az3 - a3z-1 - 2a3z + a3z5 + 2a5z-1 + 2a5z + 3a5z3 + 2a5z5 - 2a7z-1 - 2a7z + a7z5 + a9z-1 - a9z3

Kauffman Polynomial:

- az + 2az3 - az5 - 2a2z2 + 5a2z4 - 3a2z6 - a3z-1 + 4a3z - 6a3z3 + 8a3z5 - 5a3z7 - a4z2 + a4z4 + 4a4z6 - 5a4z8 - 2a5z-1 + 10a5z - 22a5z3 + 21a5z5 - 6a5z7 - 3a5z9 - a6 + 7a6z2 - 18a6z4 + 22a6z6 - 10a6z8 - a6z10 - 2a7z-1 + 12a7z - 25a7z3 + 16a7z5 + 5a7z7 - 7a7z9 + 9a8z2 - 28a8z4 + 32a8z6 - 11a8z8 - a8z10 - a9z-1 + 7a9z - 18a9z3 + 15a9z5 + 2a9z7 - 4a9z9 + 3a10z2 - 12a10z4 + 16a10z6 - 6a10z8 - 7a11z3 + 11a11z5 - 4a11z7 + 2a12z4 - a12z6

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 2                       1 j = 0                     2   j = -2                   5 1   j = -4                 7 3     j = -6               9 4       j = -8             9 7         j = -10           9 9           j = -12         7 9             j = -14       5 9               j = -16     3 8                 j = -18   1 4                   j = -20   3                     j = -22 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, 185]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 185]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 3, 11, 4], X[14, 17, 15, 18], X[16, 5, 17, 6], > X[4, 15, 5, 16], X[18, 11, 19, 12], X[22, 19, 7, 20], X[20, 14, 21, 13], > X[12, 22, 13, 21], X[2, 7, 3, 8], X[6, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -5, 4, -11}, > {10, -1, 11, -2, 6, -9, 8, -3, 5, -4, 3, -6, 7, -8, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 4 7 12 16 18 18 16 11 7 q - ----- + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q q 3 > ------- - Sqrt[q] Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 2 -28 -26 5 -22 -20 -18 3 3 -12 -1 - q + --- + q - q + --- - q - q + q - --- + --- - q + 30 24 16 14 q q q q 2 3 3 3 2 > --- + -- - -- + -- + q 10 8 6 4 q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 185]][a, z]
Out[8]=	3 5 7 9 a 2 a 2 a a 3 5 7 3 5 3 -(--) + ---- - ---- + -- - a z - 2 a z + 2 a z - 2 a z - a z + 3 a z - z z z z 9 3 3 5 5 5 7 5 > a z + a z + 2 a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 185]][a, z]
Out[9]=	3 5 7 9 6 a 2 a 2 a a 3 5 7 9 -a - -- - ---- - ---- - -- - a z + 4 a z + 10 a z + 12 a z + 7 a z - z z z z 2 2 4 2 6 2 8 2 10 2 3 3 3 > 2 a z - a z + 7 a z + 9 a z + 3 a z + 2 a z - 6 a z - 5 3 7 3 9 3 11 3 2 4 4 4 6 4 > 22 a z - 25 a z - 18 a z - 7 a z + 5 a z + a z - 18 a z - 8 4 10 4 12 4 5 3 5 5 5 7 5 > 28 a z - 12 a z + 2 a z - a z + 8 a z + 21 a z + 16 a z + 9 5 11 5 2 6 4 6 6 6 8 6 > 15 a z + 11 a z - 3 a z + 4 a z + 22 a z + 32 a z + 10 6 12 6 3 7 5 7 7 7 9 7 11 7 > 16 a z - a z - 5 a z - 6 a z + 5 a z + 2 a z - 4 a z - 4 8 6 8 8 8 10 8 5 9 7 9 9 9 > 5 a z - 10 a z - 11 a z - 6 a z - 3 a z - 7 a z - 4 a z - 6 10 8 10 > a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 5 1 3 1 4 3 8 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 22 9 20 8 18 8 18 7 16 7 16 6 14 6 q q q t q t q t q t q t q t q t 9 7 9 9 9 9 7 9 4 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 14 5 12 5 12 4 10 4 10 3 8 3 8 2 6 2 6 q t q t q t q t q t q t q t q t q t 7 t 2 2 > ---- + 2 t + -- + q t 4 2 q t q

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