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Knotscape

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DrawMorseLink

PD Presentation:

X12,1,13,2 X16,7,17,8 X3948 X17,2,18,3 X14,6,15,5 X6,12,7,11 X9,18,10,19 X20,15,11,16 X10,13,1,14 X4,19,5,20

Gauss Code:

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

Jones Polynomial:

q-15/2 - 3q-13/2 + 4q-11/2 - 6q-9/2 + 6q-7/2 - 6q-5/2 + 5q-3/2 - 4q-1/2 + q1/2

A2 (sl(3)) Invariant:

- q-24 + 2q-20 + 2q-16 + q-14 + 2q-10 + 2q-6 + 2 - q2

HOMFLY-PT Polynomial:

az3 - a3z-1 - 4a3z - 3a3z3 - a3z5 + a5z-1 + 3a5z + 2a5z3 - a7z

Kauffman Polynomial:

- z2 + az - 4az3 - 2a2z2 + 2a2z4 - 2a2z6 - a3z-1 + 8a3z - 17a3z3 + 13a3z5 - 4a3z7 + a4 - 2a4z2 + 2a4z4 + 3a4z6 - 2a4z8 - a5z-1 + 9a5z - 23a5z3 + 24a5z5 - 7a5z7 - 3a6z2 + 3a6z4 + 4a6z6 - 2a6z8 + 2a7z - 10a7z3 + 11a7z5 - 3a7z7 - 2a8z2 + 3a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 j = 2                 1 j = 0               3   j = -2             3 2   j = -4           3 2     j = -6         3 3       j = -8       3 3         j = -10     2 4           j = -12   1 2             j = -14   2               j = -16 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, NonAlternating, 64]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 64]]
Out[4]=	PD[X[12, 1, 13, 2], X[16, 7, 17, 8], X[3, 9, 4, 8], X[17, 2, 18, 3], > X[14, 6, 15, 5], X[6, 12, 7, 11], X[9, 18, 10, 19], X[20, 15, 11, 16], > X[10, 13, 1, 14], X[4, 19, 5, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 4, -3, -10, 5, -6, 2, 3, -7, -9}, > {6, -1, 9, -5, 8, -2, -4, 7, 10, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 3 4 6 6 6 5 4 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + Sqrt[q] 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 2 2 -14 2 2 2 2 - q + --- + --- + q + --- + -- - q 20 16 10 6 q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 64]][a, z]
Out[8]=	3 5 a a 3 5 7 3 3 3 5 3 3 5 -(--) + -- - 4 a z + 3 a z - a z + a z - 3 a z + 2 a z - a z z z
In[9]:=	Kauffman[Link[10, NonAlternating, 64]][a, z]
Out[9]=	3 5 4 a a 3 5 7 2 2 2 4 2 a - -- - -- + a z + 8 a z + 9 a z + 2 a z - z - 2 a z - 2 a z - z z 6 2 8 2 3 3 3 5 3 7 3 2 4 > 3 a z - 2 a z - 4 a z - 17 a z - 23 a z - 10 a z + 2 a z + 4 4 6 4 8 4 3 5 5 5 7 5 2 6 > 2 a z + 3 a z + 3 a z + 13 a z + 24 a z + 11 a z - 2 a z + 4 6 6 6 8 6 3 7 5 7 7 7 4 8 6 8 > 3 a z + 4 a z - a z - 4 a z - 7 a z - 3 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 2 2 4 3 3 3 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 3 3 3 2 3 2 > ----- + ----- + ----- + ---- + ---- + q t 6 3 6 2 4 2 4 2 q t q t q t q t q t

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L10n63 L10n65