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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X2536 X16,11,13,12 X10,3,11,4 X4,9,1,10 X14,7,15,8 X8,13,5,14 X12,15,9,16

Gauss Code:

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

Jones Polynomial:

- q-19/2 + q-17/2 - 5q-15/2 + 4q-13/2 - 7q-11/2 + 4q-9/2 - 6q-7/2 + 3q-5/2 - q-3/2

A2 (sl(3)) Invariant:

q-32 + 4q-30 + 6q-28 + 8q-26 + 13q-24 + 12q-22 + 11q-20 + 10q-18 + 6q-16 + 6q-14 + 2q-12 + 2q-10 + q-8 - 2q-6 + q-4

HOMFLY-PT Polynomial:

- a3z3 - a5z-3 - 4a5z-1 - 6a5z - 3a5z3 + 3a7z-3 + 8a7z-1 + 6a7z - 3a9z-3 - 4a9z-1 + a11z-3

Kauffman Polynomial:

- a3z3 - 3a4z4 - a5z-3 + 4a5z-1 - 6a5z + 9a5z3 - 6a5z5 + 3a6z-2 - 8a6 + 6a6z2 + 2a6z4 - 4a6z6 - 3a7z-3 + 9a7z-1 - 14a7z + 17a7z3 - 7a7z5 - a7z7 + 6a8z-2 - 15a8 + 12a8z2 + 5a8z4 - 5a8z6 - 3a9z-3 + 9a9z-1 - 14a9z + 11a9z3 - 2a9z5 - a9z7 + 3a10z-2 - 8a10 + 6a10z2 - a10z6 - a11z-3 + 4a11z-1 - 6a11z + 4a11z3 - a11z5

Khovanov Homology:

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

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