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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-25/2 + 2q-23/2 - 4q-21/2 + 6q-19/2 - 7q-17/2 + 8q-15/2 - 8q-13/2 + 5q-11/2 - 4q-9/2 + 2q-7/2 - q-5/2

A2 (sl(3)) Invariant:

q-38 + q-32 - q-30 + q-28 + 2q-22 + 3q-18 + q-16 + q-12 - q-10 + q-8

HOMFLY-PT Polynomial:

- a5z - 3a5z3 - a5z5 - a7z-1 - 6a7z - 7a7z3 - 2a7z5 + a9z-1 + a9z - 2a9z3 - a9z5 + 2a11z + a11z3

Kauffman Polynomial:

- a5z + 3a5z3 - a5z5 - a6z2 + 5a6z4 - 2a6z6 - a7z-1 + 7a7z - 12a7z3 + 10a7z5 - 3a7z7 + a8 - 3a8z2 - a8z4 + 4a8z6 - 2a8z8 - a9z-1 + 5a9z - 13a9z3 + 6a9z5 - a9z9 + 7a10z2 - 19a10z4 + 13a10z6 - 4a10z8 - a11z - a11z5 + a11z7 - a11z9 + 7a12z2 - 8a12z4 + 5a12z6 - 2a12z8 + a13z3 + 3a13z5 - 2a13z7 - 2a14z2 + 5a14z4 - 2a14z6 - 2a15z + 3a15z3 - a15z5

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 = -4                     1 j = -6                   2 1 j = -8                 2     j = -10               3 2     j = -12             5 2       j = -14           3 3         j = -16         4 5           j = -18       2 3             j = -20     2 4               j = -22   1 3                 j = -24   1                   j = -26 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, 100]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 100]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 3, 13, 4], X[14, 19, 15, 20], X[18, 7, 19, 8], > X[16, 5, 17, 6], X[4, 15, 5, 16], X[6, 17, 7, 18], X[20, 13, 9, 14], > X[2, 9, 3, 10], X[8, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -6, 5, -7, 4, -10}, > {9, -1, 10, -2, 8, -3, 6, -5, 7, -4, 3, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(25/2) 2 4 6 7 8 8 5 4 -q + ----- - ----- + ----- - ----- + ----- - ----- + ----- - ---- + 23/2 21/2 19/2 17/2 15/2 13/2 11/2 9/2 q q q q q q q q 2 -(5/2) > ---- - q 7/2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-38 -32 -30 -28 2 3 -16 -12 -10 -8 q + q - q + q + --- + --- + q + q - q + q 22 18 q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 100]][a, z]
Out[8]=	7 9 a a 5 7 9 11 5 3 7 3 9 3 -(--) + -- - a z - 6 a z + a z + 2 a z - 3 a z - 7 a z - 2 a z + z z 11 3 5 5 7 5 9 5 > a z - a z - 2 a z - a z
In[9]:=	Kauffman[Link[10, Alternating, 100]][a, z]
Out[9]=	7 9 8 a a 5 7 9 11 15 6 2 8 2 a - -- - -- - a z + 7 a z + 5 a z - a z - 2 a z - a z - 3 a z + z z 10 2 12 2 14 2 5 3 7 3 9 3 13 3 > 7 a z + 7 a z - 2 a z + 3 a z - 12 a z - 13 a z + a z + 15 3 6 4 8 4 10 4 12 4 14 4 5 5 > 3 a z + 5 a z - a z - 19 a z - 8 a z + 5 a z - a z + 7 5 9 5 11 5 13 5 15 5 6 6 8 6 > 10 a z + 6 a z - a z + 3 a z - a z - 2 a z + 4 a z + 10 6 12 6 14 6 7 7 11 7 13 7 8 8 > 13 a z + 5 a z - 2 a z - 3 a z + a z - 2 a z - 2 a z - 10 8 12 8 9 9 11 9 > 4 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 1 1 3 2 4 2 q + q + ------- + ------ + ------ + ------ + ------ + ------ + ------ + 26 10 24 9 22 9 22 8 20 8 20 7 18 7 q t q t q t q t q t q t q t 3 4 5 3 3 5 2 3 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 18 6 16 6 16 5 14 5 14 4 12 4 12 3 10 3 q t q t q t q t q t q t q t q t 2 2 2 > ------ + ----- + ---- 10 2 8 2 6 q t q t q t

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