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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X14,8,15,7 X15,20,16,21 X11,18,12,19 X19,12,20,13 X17,22,18,5 X21,16,22,17 X8,14,9,13 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-21/2 - q-19/2 + q-17/2 - 2q-13/2 + q-11/2 - 3q-9/2 + 3q-7/2 - 3q-5/2 + 2q-3/2 - q-1/2

A2 (sl(3)) Invariant:

- q-34 - 2q-32 - 2q-28 + q-24 + 2q-22 + 4q-20 + 2q-18 + 4q-16 + q-2

HOMFLY-PT Polynomial:

- 3a3z - 4a3z3 - a3z5 - a5z-1 + 2a5z + 6a5z3 + 5a5z5 + a5z7 - 4a7z - 5a7z3 - a7z5 + 2a9z-1 + 3a9z - a11z-1

Kauffman Polynomial:

3a3z - 7a3z3 + 5a3z5 - a3z7 + 5a4z2 - 13a4z4 + 10a4z6 - 2a4z8 + a5z-1 - a5z - 6a5z3 + 3a5z5 + 3a5z7 - a5z9 - a6 + 10a6z2 - 22a6z4 + 16a6z6 - 3a6z8 - 3a7z + 3a7z3 - 5a7z5 + 5a7z7 - a7z9 + 3a8 - a8z2 - 6a8z4 + 6a8z6 - a8z8 - 2a9z-1 + 3a9z - 3a9z3 + 2a9z5 + 5a10 - 12a10z2 + 8a10z4 - a10z6 - a11z-1 + 2a11z - 5a11z3 + 5a11z5 - a11z7 + 2a12 - 6a12z2 + 5a12z4 - 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 = 0                       1 j = -2                     1   j = -4                   2 1   j = -6                 2 2     j = -8             1 2 1       j = -10           1 1 2         j = -12         1 4 2           j = -14       1   1             j = -16       1 2               j = -18   1 1                   j = -20                         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, NonAlternating, 64]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 64]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[14, 8, 15, 7], X[15, 20, 16, 21], > X[11, 18, 12, 19], X[19, 12, 20, 13], X[17, 22, 18, 5], X[21, 16, 22, 17], > X[8, 14, 9, 13], 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, 6, 9, -3, -4, 8, -7, 5, > -6, 4, -8, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) -(19/2) -(17/2) 2 -(11/2) 3 3 3 2 q - q + q - ----- + q - ---- + ---- - ---- + ---- - 13/2 9/2 7/2 5/2 3/2 q q q q q 1 > ------- Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 2 2 -24 2 4 2 4 -2 -q - --- - --- + q + --- + --- + --- + --- + q 32 28 22 20 18 16 q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 64]][a, z]
Out[8]=	5 9 11 a 2 a a 3 5 7 9 3 3 5 3 -(--) + ---- - --- - 3 a z + 2 a z - 4 a z + 3 a z - 4 a z + 6 a z - z z z 7 3 3 5 5 5 7 5 5 7 > 5 a z - a z + 5 a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 64]][a, z]
Out[9]=	5 9 11 6 8 10 12 a 2 a a 3 5 7 -a + 3 a + 5 a + 2 a + -- - ---- - --- + 3 a z - a z - 3 a z + z z z 9 11 4 2 6 2 8 2 10 2 12 2 > 3 a z + 2 a z + 5 a z + 10 a z - a z - 12 a z - 6 a z - 3 3 5 3 7 3 9 3 11 3 4 4 6 4 > 7 a z - 6 a z + 3 a z - 3 a z - 5 a z - 13 a z - 22 a z - 8 4 10 4 12 4 3 5 5 5 7 5 9 5 > 6 a z + 8 a z + 5 a z + 5 a z + 3 a z - 5 a z + 2 a z + 11 5 4 6 6 6 8 6 10 6 12 6 3 7 > 5 a z + 10 a z + 16 a z + 6 a z - a z - a z - a z + 5 7 7 7 11 7 4 8 6 8 8 8 5 9 7 9 > 3 a z + 5 a z - 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 1 1 2 1 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 6 4 22 9 18 8 18 7 16 6 14 6 16 5 12 5 q q q t q t q t q t q t q t q t 1 4 1 2 1 1 2 2 > ------ + ------ + ------ + ------ + ------ + ----- + ------ + ----- + 14 4 12 4 10 4 12 3 10 3 8 3 10 2 8 2 q t q t q t q t q t q t q t q t 1 2 t t 2 > ---- + ---- + -- + -- + t 8 6 4 2 q t q t q q

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