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Knotscape

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DrawMorseLink

PD Presentation:

X8192 X20,9,21,10 X21,1,22,6 X7,18,8,19 X3,10,4,11 X15,12,16,13 X5,14,6,15 X13,4,14,5 X11,16,12,17 X17,22,18,7 X2,20,3,19

Gauss Code:

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

Jones Polynomial:

q-21/2 - 3q-19/2 + 5q-17/2 - 7q-15/2 + 9q-13/2 - 9q-11/2 + 8q-9/2 - 7q-7/2 + 3q-5/2 - 2q-3/2

A2 (sl(3)) Invariant:

- q-32 + q-30 - q-26 + 2q-24 - 2q-22 - q-20 - q-18 - q-16 + 3q-14 + q-12 + 4q-10 + 3q-8 + 2q-4

HOMFLY-PT Polynomial:

- 2a3z-1 - 4a3z - 2a3z3 + 3a5z-1 + 3a5z + 2a5z3 + a5z5 - a7z-1 + a7z + 2a7z3 + a7z5 - a9z - a9z3

Kauffman Polynomial:

- 2a3z-1 + 5a3z - 3a3z3 + 3a4 - 3a4z2 - a4z6 - 3a5z-1 + 7a5z - 8a5z3 + 3a5z5 - 2a5z7 + 3a6 - 4a6z2 - a6z4 + 2a6z6 - 2a6z8 - a7z-1 + 2a7z - 6a7z3 + 7a7z5 - 2a7z7 - a7z9 + a8 + 2a8z2 - 8a8z4 + 12a8z6 - 5a8z8 + a9z - 9a9z3 + 14a9z5 - 3a9z7 - a9z9 + a10z2 - 4a10z4 + 8a10z6 - 3a10z8 + a11z - 8a11z3 + 10a11z5 - 3a11z7 - 2a12z2 + 3a12z4 - a12z6

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -2                   2 j = -4                 2 1 j = -6               5 1   j = -8             4 3     j = -10           5 4       j = -12         4 4         j = -14       3 5           j = -16     2 4             j = -18   1 3               j = -20   2                 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, 183]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 183]]
Out[4]=	PD[X[8, 1, 9, 2], X[20, 9, 21, 10], X[21, 1, 22, 6], X[7, 18, 8, 19], > X[3, 10, 4, 11], X[15, 12, 16, 13], X[5, 14, 6, 15], X[13, 4, 14, 5], > X[11, 16, 12, 17], X[17, 22, 18, 7], X[2, 20, 3, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, -5, 8, -7, 3}, > {-4, -1, 2, 5, -9, 6, -8, 7, -6, 9, -10, 4, 11, -2, -3, 10}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 3 5 7 9 9 8 7 3 2 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
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 -30 -26 2 2 -20 -18 -16 3 -12 4 3 -q + q - q + --- - --- - q - q - q + --- + q + --- + -- + 24 22 14 10 8 q q q q q 2 > -- 4 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 183]][a, z]
Out[8]=	3 5 7 -2 a 3 a a 3 5 7 9 3 3 5 3 ----- + ---- - -- - 4 a z + 3 a z + a z - a z - 2 a z + 2 a z + z z z 7 3 9 3 5 5 7 5 > 2 a z - a z + a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 183]][a, z]
Out[9]=	3 5 7 4 6 8 2 a 3 a a 3 5 7 9 11 3 a + 3 a + a - ---- - ---- - -- + 5 a z + 7 a z + 2 a z + a z + a z - z z z 4 2 6 2 8 2 10 2 12 2 3 3 5 3 > 3 a z - 4 a z + 2 a z + a z - 2 a z - 3 a z - 8 a z - 7 3 9 3 11 3 6 4 8 4 10 4 12 4 > 6 a z - 9 a z - 8 a z - a z - 8 a z - 4 a z + 3 a z + 5 5 7 5 9 5 11 5 4 6 6 6 8 6 > 3 a z + 7 a z + 14 a z + 10 a z - a z + 2 a z + 12 a z + 10 6 12 6 5 7 7 7 9 7 11 7 6 8 > 8 a z - a z - 2 a z - 2 a z - 3 a z - 3 a z - 2 a z - 8 8 10 8 7 9 9 9 > 5 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 2 1 2 1 3 2 4 3 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 22 9 20 8 18 8 18 7 16 7 16 6 14 6 q q t q t q t q t q t q t q t 5 4 4 5 4 4 3 5 1 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 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 2 > ---- 4 q t

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