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Knotscape

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DrawMorseLink

PD Presentation:

X10,1,11,2 X2,11,3,12 X12,3,13,4 X16,10,17,9 X6,13,7,14 X14,7,15,8 X8,15,1,16 X20,18,9,17 X4,19,5,20 X18,5,19,6

Gauss Code:

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

Jones Polynomial:

- q-21/2 + 2q-19/2 - 4q-17/2 + 6q-15/2 - 8q-13/2 + 7q-11/2 - 7q-9/2 + 6q-7/2 - 4q-5/2 + 2q-3/2 - q-1/2

A2 (sl(3)) Invariant:

q-32 + q-30 + q-26 - q-24 + 2q-22 + 2q-20 + 2q-18 + 3q-16 - 2q-14 - 2q-10 + q-6 + q-2

HOMFLY-PT Polynomial:

- 4a3z - 4a3z3 - a3z5 + 5a5z + 8a5z3 + 5a5z5 + a5z7 - a7z-1 - 8a7z - 8a7z3 - 2a7z5 + a9z-1 + 3a9z + a9z3

Kauffman Polynomial:

4a3z - 8a3z3 + 5a3z5 - a3z7 + 5a4z2 - 12a4z4 + 9a4z6 - 2a4z8 + 4a5z - 16a5z3 + 12a5z5 - a5z9 + 2a6z2 - 13a6z4 + 17a6z6 - 5a6z8 - a7z-1 + 9a7z - 25a7z3 + 23a7z5 - 4a7z7 - a7z9 + a8 - 6a8z2 + 6a8z4 + 4a8z6 - 3a8z8 - a9z-1 + 7a9z - 14a9z3 + 13a9z5 - 5a9z7 - 2a10z2 + 5a10z4 - 4a10z6 - a11z + 2a11z3 - 3a11z5 + a12z2 - 2a12z4 + a13z - a13z3

Khovanov Homology:

trqj 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                 3 1   j = -6               4 2     j = -8             3 2       j = -10           4 4         j = -12         4 3           j = -14       2 4             j = -16     2 4               j = -18     2                 j = -20 1 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[10, Alternating, 105]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 105]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[16, 10, 17, 9], > X[6, 13, 7, 14], X[14, 7, 15, 8], X[8, 15, 1, 16], X[20, 18, 9, 17], > X[4, 19, 5, 20], X[18, 5, 19, 6]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -9, 10, -5, 6, -7}, > {4, -1, 2, -3, 5, -6, 7, -4, 8, -10, 9, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 2 4 6 8 7 7 6 4 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 1 > ------- Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 -30 -26 -24 2 2 2 3 2 2 -6 -2 q + q + q - q + --- + --- + --- + --- - --- - --- + q + q 22 20 18 16 14 10 q q q q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 105]][a, z]
Out[8]=	7 9 a a 3 5 7 9 3 3 5 3 7 3 -(--) + -- - 4 a z + 5 a z - 8 a z + 3 a z - 4 a z + 8 a z - 8 a z + z z 9 3 3 5 5 5 7 5 5 7 > a z - a z + 5 a z - 2 a z + a z
In[9]:=	Kauffman[Link[10, Alternating, 105]][a, z]
Out[9]=	7 9 8 a a 3 5 7 9 11 13 4 2 a - -- - -- + 4 a z + 4 a z + 9 a z + 7 a z - a z + a z + 5 a z + z z 6 2 8 2 10 2 12 2 3 3 5 3 7 3 > 2 a z - 6 a z - 2 a z + a z - 8 a z - 16 a z - 25 a z - 9 3 11 3 13 3 4 4 6 4 8 4 10 4 > 14 a z + 2 a z - a z - 12 a z - 13 a z + 6 a z + 5 a z - 12 4 3 5 5 5 7 5 9 5 11 5 4 6 > 2 a z + 5 a z + 12 a z + 23 a z + 13 a z - 3 a z + 9 a z + 6 6 8 6 10 6 3 7 7 7 9 7 4 8 > 17 a z + 4 a z - 4 a z - a z - 4 a z - 5 a z - 2 a z - 6 8 8 8 5 9 7 9 > 5 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 3 1 1 2 2 2 4 2 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 6 4 22 8 20 8 20 7 18 6 16 6 16 5 14 5 q q q t q t q t q t q t q t q t 4 4 3 4 4 3 2 4 t > ------ + ------ + ------ + ------ + ------ + ----- + ---- + ---- + -- + 14 4 12 4 12 3 10 3 10 2 8 2 8 6 4 q t q t q t q t q t q t q t q t q t 2 > -- + t 2 q

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