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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X10,5,11,6 X14,3,15,4 X20,11,5,12 X18,13,19,14 X12,19,13,20 X2,9,3,10 X8,15,9,16

Gauss Code:

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

Jones Polynomial:

- q-23/2 + 3q-21/2 - 4q-19/2 + 7q-17/2 - 9q-15/2 + 9q-13/2 - 10q-11/2 + 7q-9/2 - 6q-7/2 + 3q-5/2 - q-3/2

A2 (sl(3)) Invariant:

q-36 - q-34 - 3q-32 - 2q-28 - q-26 + 3q-24 + q-22 + 3q-20 + 2q-18 + 2q-16 + 3q-14 - q-12 + 2q-10 + q-8 - 2q-6 + q-4

HOMFLY-PT Polynomial:

- a3z3 - a5z-1 - 3a5z - 3a5z3 - 2a7z - 3a7z3 + 2a9z-1 + 2a9z - a9z3 - a11z-1 + a11z

Kauffman Polynomial:

- a3z3 - 3a4z4 + a5z-1 - 3a5z + 6a5z3 - 6a5z5 - a6 + 8a6z4 - 7a6z6 - 3a7z3 + 12a7z5 - 7a7z7 + 3a8 - 4a8z2 + a8z4 + 9a8z6 - 5a8z8 - 2a9z-1 + 4a9z - 15a9z3 + 17a9z5 - a9z7 - 2a9z9 + 5a10 + 2a10z2 - 29a10z4 + 30a10z6 - 8a10z8 - a11z-1 + 2a11z - 9a11z3 + 3a11z5 + 5a11z7 - 2a11z9 + 2a12 + 6a12z2 - 19a12z4 + 14a12z6 - 3a12z8 + a13z - 4a13z3 + 4a13z5 - a13z7

Khovanov Homology:

trqj r = -10 r = -9 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               4 3     j = -10             6 3       j = -12           4 5         j = -14         5 5           j = -16       2 4             j = -18     2 5               j = -20   1 2                 j = -22   2                   j = -24 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[10, Alternating, 50]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 50]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[10, 5, 11, 6], > X[14, 3, 15, 4], X[20, 11, 5, 12], X[18, 13, 19, 14], X[12, 19, 13, 20], > X[2, 9, 3, 10], X[8, 15, 9, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 5, -3}, {4, -1, 2, -10, 9, -4, 6, -8, 7, -5, 10, -2, 3, -7, > 8, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(23/2) 3 4 7 9 9 10 7 6 -q + ----- - ----- + ----- - ----- + ----- - ----- + ---- - ---- + 21/2 19/2 17/2 15/2 13/2 11/2 9/2 7/2 q q q q q q q q 3 -(3/2) > ---- - q 5/2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-36 -34 3 2 -26 3 -22 3 2 2 3 -12 q - q - --- - --- - q + --- + q + --- + --- + --- + --- - q + 32 28 24 20 18 16 14 q q q q q q q 2 -8 2 -4 > --- + q - -- + q 10 6 q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 50]][a, z]
Out[8]=	5 9 11 a 2 a a 5 7 9 11 3 3 5 3 -(--) + ---- - --- - 3 a z - 2 a z + 2 a z + a z - a z - 3 a z - z z z 7 3 9 3 > 3 a z - a z
In[9]:=	Kauffman[Link[10, Alternating, 50]][a, z]
Out[9]=	5 9 11 6 8 10 12 a 2 a a 5 9 11 -a + 3 a + 5 a + 2 a + -- - ---- - --- - 3 a z + 4 a z + 2 a z + z z z 13 8 2 10 2 12 2 3 3 5 3 7 3 > a z - 4 a z + 2 a z + 6 a z - a z + 6 a z - 3 a z - 9 3 11 3 13 3 4 4 6 4 8 4 10 4 > 15 a z - 9 a z - 4 a z - 3 a z + 8 a z + a z - 29 a z - 12 4 5 5 7 5 9 5 11 5 13 5 6 6 > 19 a z - 6 a z + 12 a z + 17 a z + 3 a z + 4 a z - 7 a z + 8 6 10 6 12 6 7 7 9 7 11 7 13 7 > 9 a z + 30 a z + 14 a z - 7 a z - a z + 5 a z - a z - 8 8 10 8 12 8 9 9 11 9 > 5 a z - 8 a z - 3 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 -2 1 2 1 2 2 5 2 q + q + ------- + ------ + ------ + ------ + ------ + ------ + ------ + 24 10 22 9 20 9 20 8 18 8 18 7 16 7 q t q t q t q t q t q t q t 4 5 5 4 5 6 3 4 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 16 6 14 6 14 5 12 5 12 4 10 4 10 3 8 3 q t q t q t q t q t q t q t q t 3 3 3 > ----- + ----- + ---- 8 2 6 2 4 q t q t q t

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