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Knotscape

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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X18,11,19,12 X20,16,7,15 X16,20,17,19 X12,17,13,18 X6718 X4,13,5,14

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

q-32 + q-30 + q-26 - q-24 + q-22 + 2q-16 - q-14 + 2q-12 + q-8 + q-6 + q-2

HOMFLY-PT Polynomial:

- a3z-1 - 4a3z - 4a3z3 - a3z5 + 2a5z-1 + 6a5z + 8a5z3 + 5a5z5 + a5z7 - 2a7z-1 - 8a7z - 8a7z3 - 2a7z5 + a9z-1 + 3a9z + a9z3

Kauffman Polynomial:

- a3z-1 + 5a3z - 8a3z3 + 5a3z5 - a3z7 + 3a4z2 - 11a4z4 + 9a4z6 - 2a4z8 - 2a5z-1 + 11a5z - 23a5z3 + 14a5z5 - a5z9 - a6 + 8a6z2 - 20a6z4 + 19a6z6 - 5a6z8 - 2a7z-1 + 12a7z - 24a7z3 + 20a7z5 - 3a7z7 - a7z9 - a8z4 + 6a8z6 - 3a8z8 - a9z-1 + 4a9z - 6a9z3 + 8a9z5 - 4a9z7 - 4a10z2 + 6a10z4 - 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               3 2     j = -8             4 2       j = -10           3 3         j = -12         4 4           j = -14       2 3             j = -16     2 4               j = -18   1 3                 j = -20   1                   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, 67]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 67]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[14, 5, 15, 6], > X[18, 11, 19, 12], X[20, 16, 7, 15], X[16, 20, 17, 19], X[12, 17, 13, 18], > X[6, 7, 1, 8], X[4, 13, 5, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -10, 4, -9}, > {9, -1, 2, -3, 5, -8, 10, -4, 6, -7, 8, -5, 7, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 2 4 6 7 7 7 5 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 -22 2 -14 2 -8 -6 -2 q + q + q - q + q + --- - q + --- + q + q + q 16 12 q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 67]][a, z]
Out[8]=	3 5 7 9 a 2 a 2 a a 3 5 7 9 3 3 -(--) + ---- - ---- + -- - 4 a z + 6 a z - 8 a z + 3 a z - 4 a z + z z z z 5 3 7 3 9 3 3 5 5 5 7 5 5 7 > 8 a z - 8 a z + a z - a z + 5 a z - 2 a z + a z
In[9]:=	Kauffman[Link[10, Alternating, 67]][a, z]
Out[9]=	3 5 7 9 6 a 2 a 2 a a 3 5 7 9 11 -a - -- - ---- - ---- - -- + 5 a z + 11 a z + 12 a z + 4 a z - a z + z z z z 13 4 2 6 2 10 2 12 2 3 3 5 3 > a z + 3 a z + 8 a z - 4 a z + a z - 8 a z - 23 a z - 7 3 9 3 11 3 13 3 4 4 6 4 8 4 > 24 a z - 6 a z + 2 a z - a z - 11 a z - 20 a z - a z + 10 4 12 4 3 5 5 5 7 5 9 5 11 5 > 6 a z - 2 a z + 5 a z + 14 a z + 20 a z + 8 a z - 3 a z + 4 6 6 6 8 6 10 6 3 7 7 7 9 7 > 9 a z + 19 a z + 6 a z - 4 a z - a z - 3 a z - 4 a z - 4 8 6 8 8 8 5 9 7 9 > 2 a z - 5 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 3 1 1 1 3 2 4 2 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 6 4 22 8 20 7 18 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 3 4 4 3 3 4 2 3 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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