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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-25/2 + 3q-23/2 - 6q-21/2 + 9q-19/2 - 11q-17/2 + 12q-15/2 - 11q-13/2 + 8q-11/2 - 6q-9/2 + 2q-7/2 - q-5/2

A2 (sl(3)) Invariant:

q-38 - q-36 + 2q-32 - 2q-30 + q-28 - q-26 - q-24 + 2q-22 - q-20 + 4q-18 + q-16 + q-14 + 3q-12 - q-10 + q-8

HOMFLY-PT Polynomial:

- a5z-1 - 3a5z - 3a5z3 - a5z5 + a7z-1 - 2a7z - 5a7z3 - 2a7z5 + a9z - a9z3 - a9z5 + a11z + a11z3

Kauffman Polynomial:

a5z-1 - 3a5z + 3a5z3 - a5z5 - a6 + 3a6z4 - 2a6z6 + a7z-1 - a7z - a7z3 + 4a7z5 - 3a7z7 - 6a8z2 + 10a8z4 - 3a8z6 - 2a8z8 - a9z + 2a9z3 + 4a9z5 - 4a9z7 - a9z9 - 3a10z2 + 4a10z4 + 3a10z6 - 5a10z8 - 3a11z + 3a11z3 + 6a11z5 - 5a11z7 - a11z9 + 3a12z4 + a12z6 - 3a12z8 - a13z - a13z3 + 6a13z5 - 4a13z7 - 3a14z2 + 6a14z4 - 3a14z6 - a15z + 2a15z3 - 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                 4     j = -10               4 2     j = -12             7 4       j = -14           5 4         j = -16         6 7           j = -18       4 6             j = -20     2 5               j = -22   1 4                 j = -24   2                   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, 87]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 87]]
Out[4]=	PD[X[8, 1, 9, 2], X[20, 9, 7, 10], X[6, 7, 1, 8], X[2, 11, 3, 12], > X[12, 3, 13, 4], X[16, 5, 17, 6], X[18, 13, 19, 14], X[14, 17, 15, 18], > X[4, 15, 5, 16], X[10, 19, 11, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 5, -9, 6, -3}, > {3, -1, 2, -10, 4, -5, 7, -8, 9, -6, 8, -7, 10, -2}]
In[6]:=	Jones[L][q]
Out[6]=	-(25/2) 3 6 9 11 12 11 8 6 -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 -36 2 2 -28 -26 -24 2 -20 4 -16 -14 q - q + --- - --- + q - q - q + --- - q + --- + q + q + 32 30 22 18 q q q q 3 -10 -8 > --- - q + q 12 q
In[8]:=	HOMFLYPT[Link[10, Alternating, 87]][a, z]
Out[8]=	5 7 a a 5 7 9 11 5 3 7 3 9 3 -(--) + -- - 3 a z - 2 a z + a z + a z - 3 a z - 5 a z - 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, 87]][a, z]
Out[9]=	5 7 6 a a 5 7 9 11 13 15 8 2 -a + -- + -- - 3 a z - a z - a z - 3 a z - a z - a z - 6 a z - z z 10 2 14 2 5 3 7 3 9 3 11 3 13 3 > 3 a z - 3 a z + 3 a z - a z + 2 a z + 3 a z - a z + 15 3 6 4 8 4 10 4 12 4 14 4 5 5 > 2 a z + 3 a z + 10 a z + 4 a z + 3 a z + 6 a z - a z + 7 5 9 5 11 5 13 5 15 5 6 6 8 6 > 4 a z + 4 a z + 6 a z + 6 a z - a z - 2 a z - 3 a z + 10 6 12 6 14 6 7 7 9 7 11 7 13 7 > 3 a z + a z - 3 a z - 3 a z - 4 a z - 5 a z - 4 a z - 8 8 10 8 12 8 9 9 11 9 > 2 a z - 5 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 2 1 4 2 5 4 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 6 6 7 5 4 7 4 4 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 4 2 > ------ + ----- + ---- 10 2 8 2 6 q t q t q t

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