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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-27/2 + 3q-25/2 - 5q-23/2 + 7q-21/2 - 8q-19/2 + 8q-17/2 - 7q-15/2 + 5q-13/2 - 4q-11/2 + q-9/2 - q-7/2

A2 (sl(3)) Invariant:

q-40 - q-38 - q-34 - q-32 + q-30 - 2q-28 + 2q-26 + 2q-22 + 3q-20 + q-18 + 3q-16 + q-12

HOMFLY-PT Polynomial:

- 2a7z-1 - 9a7z - 12a7z3 - 6a7z5 - a7z7 + 3a9z-1 + 5a9z - 2a9z3 - 4a9z5 - a9z7 - a11z-1 + a11z + 3a11z3 + a11z5

Kauffman Polynomial:

- 2a7z-1 + 9a7z - 12a7z3 + 6a7z5 - a7z7 + 3a8 - 7a8z2 + a8z4 + 3a8z6 - a8z8 - 3a9z-1 + 9a9z - 11a9z3 + 4a9z5 + 2a9z7 - a9z9 + 3a10 - 6a10z2 - 5a10z4 + 12a10z6 - 4a10z8 - a11z-1 - 4a11z3 + 10a11z5 - 2a11z7 - a11z9 + a12 - 3a12z2 + 5a12z4 + 3a12z6 - 3a12z8 - a13z + 7a13z5 - 5a13z7 - 3a14z2 + 8a14z4 - 6a14z6 - a15z + 4a15z3 - 5a15z5 + a16z2 - 3a16z4 - a17z3

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 = -6                     1 j = -8                   1 1 j = -10                 3     j = -12               2 1     j = -14             5 3       j = -16           3 2         j = -18         5 5           j = -20       3 4             j = -22     2 4               j = -24   1 3                 j = -26   2                   j = -28 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, 74]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 74]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[16, 5, 17, 6], > X[18, 11, 19, 12], X[20, 13, 7, 14], X[12, 19, 13, 20], X[14, 17, 15, 18], > X[6, 7, 1, 8], X[4, 15, 5, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -10, 4, -9}, > {9, -1, 2, -3, 5, -7, 6, -8, 10, -4, 8, -5, 7, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(27/2) 3 5 7 8 8 7 5 4 -q + ----- - ----- + ----- - ----- + ----- - ----- + ----- - ----- + 25/2 23/2 21/2 19/2 17/2 15/2 13/2 11/2 q q q q q q q q -(9/2) -(7/2) > q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-40 -38 -34 -32 -30 2 2 2 3 -18 3 -12 q - q - q - q + q - --- + --- + --- + --- + q + --- + q 28 26 22 20 16 q q q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 74]][a, z]
Out[8]=	7 9 11 -2 a 3 a a 7 9 11 7 3 9 3 11 3 ----- + ---- - --- - 9 a z + 5 a z + a z - 12 a z - 2 a z + 3 a z - z z z 7 5 9 5 11 5 7 7 9 7 > 6 a z - 4 a z + a z - a z - a z
In[9]:=	Kauffman[Link[10, Alternating, 74]][a, z]
Out[9]=	7 9 11 8 10 12 2 a 3 a a 7 9 13 15 3 a + 3 a + a - ---- - ---- - --- + 9 a z + 9 a z - a z - a z - z z z 8 2 10 2 12 2 14 2 16 2 7 3 9 3 > 7 a z - 6 a z - 3 a z - 3 a z + a z - 12 a z - 11 a z - 11 3 15 3 17 3 8 4 10 4 12 4 14 4 > 4 a z + 4 a z - a z + a z - 5 a z + 5 a z + 8 a z - 16 4 7 5 9 5 11 5 13 5 15 5 8 6 > 3 a z + 6 a z + 4 a z + 10 a z + 7 a z - 5 a z + 3 a z + 10 6 12 6 14 6 7 7 9 7 11 7 13 7 > 12 a z + 3 a z - 6 a z - a z + 2 a z - 2 a z - 5 a z - 8 8 10 8 12 8 9 9 11 9 > a z - 4 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-8 -6 1 2 1 3 2 4 3 q + q + ------- + ------ + ------ + ------ + ------ + ------ + ------ + 28 10 26 9 24 9 24 8 22 8 22 7 20 7 q t q t q t q t q t q t q t 4 5 5 3 2 5 3 2 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 20 6 18 6 18 5 16 5 16 4 14 4 14 3 12 3 q t q t q t q t q t q t q t q t 1 3 1 > ------ + ------ + ---- 12 2 10 2 8 q t q t q t

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