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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 X20,11,7,12 X18,13,19,14 X14,17,15,18 X12,19,13,20 X6718 X4,15,5,16

Gauss Code:

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

Jones Polynomial:

- q-25/2 + 2q-23/2 - 3q-21/2 + 5q-19/2 - 6q-17/2 + 6q-15/2 - 6q-13/2 + 4q-11/2 - 3q-9/2 + q-7/2 - q-5/2

A2 (sl(3)) Invariant:

q-38 - 2q-30 + q-24 + 2q-22 + 2q-18 + q-16 + q-14 + 2q-12 + q-8

HOMFLY-PT Polynomial:

- a5z-1 - 4a5z - 4a5z3 - a5z5 + a7z-1 + a7z - 2a7z3 - a7z5 - 2a9z - 3a9z3 - a9z5 + 2a11z + a11z3

Kauffman Polynomial:

a5z-1 - 4a5z + 4a5z3 - a5z5 - a6 + a6z2 + 2a6z4 - a6z6 + a7z-1 - 3a7z + 3a7z3 + a7z5 - a7z7 + a8z2 - 2a8z4 + 2a8z6 - a8z8 - a9z + 6a9z3 - 9a9z5 + 4a9z7 - a9z9 + 8a10z2 - 20a10z4 + 12a10z6 - 3a10z8 - a11z + 2a11z3 - 5a11z5 + 3a11z7 - a11z9 + 5a12z2 - 10a12z4 + 7a12z6 - 2a12z8 - 2a13z3 + 5a13z5 - 2a13z7 - 3a14z2 + 6a14z4 - 2a14z6 - a15z + 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                   1 1 j = -8                 2     j = -10               2 1     j = -12             4 2       j = -14           2 2         j = -16         4 4           j = -18       2 3             j = -20     1 3               j = -22   1 2                 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, 75]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 75]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[16, 5, 17, 6], > X[20, 11, 7, 12], X[18, 13, 19, 14], X[14, 17, 15, 18], X[12, 19, 13, 20], > 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, -8, 6, -7, 10, -4, 7, -6, 8, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(25/2) 2 3 5 6 6 6 4 3 -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 -(7/2) -(5/2) > q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-38 2 -24 2 2 -16 -14 2 -8 q - --- + q + --- + --- + q + q + --- + q 30 22 18 12 q q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 75]][a, z]
Out[8]=	5 7 a a 5 7 9 11 5 3 7 3 9 3 -(--) + -- - 4 a z + a z - 2 a z + 2 a z - 4 a z - 2 a z - 3 a z + z z 11 3 5 5 7 5 9 5 > a z - a z - a z - a z
In[9]:=	Kauffman[Link[10, Alternating, 75]][a, z]
Out[9]=	5 7 6 a a 5 7 9 11 15 6 2 8 2 -a + -- + -- - 4 a z - 3 a z - a z - a z - a z + a z + a z + z z 10 2 12 2 14 2 5 3 7 3 9 3 11 3 > 8 a z + 5 a z - 3 a z + 4 a z + 3 a z + 6 a z + 2 a z - 13 3 15 3 6 4 8 4 10 4 12 4 > 2 a z + 3 a z + 2 a z - 2 a z - 20 a z - 10 a z + 14 4 5 5 7 5 9 5 11 5 13 5 15 5 6 6 > 6 a z - a z + a z - 9 a z - 5 a z + 5 a z - a z - a z + 8 6 10 6 12 6 14 6 7 7 9 7 11 7 > 2 a z + 12 a z + 7 a z - 2 a z - a z + 4 a z + 3 a z - 13 7 8 8 10 8 12 8 9 9 11 9 > 2 a z - a z - 3 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 1 1 2 1 3 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 4 2 2 4 2 2 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 1 2 1 > ------ + ----- + ---- 10 2 8 2 6 q t q t q t

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