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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X16,8,17,7 X17,22,18,5 X11,18,12,19 X13,20,14,21 X19,12,20,13 X21,14,22,15 X8,16,9,15 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-19/2 - 2q-17/2 + 3q-15/2 - 4q-13/2 + 4q-11/2 - 5q-9/2 + 3q-7/2 - 3q-5/2 + 2q-3/2 - q-1/2

A2 (sl(3)) Invariant:

- 2q-28 - q-26 - q-24 + q-22 + 2q-20 + 2q-18 + 4q-16 + q-14 + 2q-12 + q-2

HOMFLY-PT Polynomial:

- 3a3z - 4a3z3 - a3z5 - 2a5z-1 + 6a5z3 + 5a5z5 + a5z7 + 3a7z-1 + a7z - 3a7z3 - a7z5 - a9z-1

Kauffman Polynomial:

3a3z - 7a3z3 + 5a3z5 - a3z7 + 3a4z2 - 13a4z4 + 10a4z6 - 2a4z8 + 2a5z-1 - 3a5z - 3a5z3 + a5z5 + 3a5z7 - a5z9 - 3a6 + 9a6z2 - 16a6z4 + 13a6z6 - 3a6z8 + 3a7z-1 - 10a7z + 14a7z3 - 8a7z5 + 4a7z7 - a7z9 - 3a8 + 10a8z2 - 6a8z4 + 3a8z6 - a8z8 + a9z-1 - 4a9z + 8a9z3 - 4a9z5 - a10 + 3a10z2 - 3a10z4 - 2a11z3 - a12z2

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 0                   1 j = -2                 1   j = -4               2 1   j = -6             2 2     j = -8           3 1       j = -10         1 2         j = -12       3 3           j = -14     1 2             j = -16   1 2               j = -18   1                 j = -20 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, 70]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 70]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[16, 8, 17, 7], X[17, 22, 18, 5], > X[11, 18, 12, 19], X[13, 20, 14, 21], X[19, 12, 20, 13], X[21, 14, 22, 15], > X[8, 16, 9, 15], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, -5, 7, -6, 8, 9, -3, -4, 5, > -7, 6, -8, 4}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 2 3 4 4 5 3 3 2 1 q - ----- + ----- - ----- + ----- - ---- + ---- - ---- + ---- - ------- 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-2 -26 -24 -22 2 2 4 -14 2 -2 --- - q - q + q + --- + --- + --- + q + --- + q 28 20 18 16 12 q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 70]][a, z]
Out[8]=	5 7 9 -2 a 3 a a 3 7 3 3 5 3 7 3 3 5 ----- + ---- - -- - 3 a z + a z - 4 a z + 6 a z - 3 a z - a z + z z z 5 5 7 5 5 7 > 5 a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 70]][a, z]
Out[9]=	5 7 9 6 8 10 2 a 3 a a 3 5 7 9 -3 a - 3 a - a + ---- + ---- + -- + 3 a z - 3 a z - 10 a z - 4 a z + z z z 4 2 6 2 8 2 10 2 12 2 3 3 5 3 > 3 a z + 9 a z + 10 a z + 3 a z - a z - 7 a z - 3 a z + 7 3 9 3 11 3 4 4 6 4 8 4 10 4 > 14 a z + 8 a z - 2 a z - 13 a z - 16 a z - 6 a z - 3 a z + 3 5 5 5 7 5 9 5 4 6 6 6 8 6 > 5 a z + a z - 8 a z - 4 a z + 10 a z + 13 a z + 3 a z - 3 7 5 7 7 7 4 8 6 8 8 8 5 9 7 9 > a z + 3 a z + 4 a z - 2 a z - 3 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 1 1 2 1 2 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 6 4 20 7 18 6 16 6 16 5 14 5 14 4 12 4 q q q t q t q t q t q t q t q t 3 1 2 3 1 2 t t 2 > ------ + ------ + ------ + ----- + ---- + ---- + -- + -- + t 12 3 10 3 10 2 8 2 8 6 4 2 q t q t q t q t q t q t q q

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L11n69 L11n71