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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X5,12,6,13 X3849 X13,2,14,3 X14,7,15,8 X9,18,10,19 X11,16,12,17 X17,20,18,11 X4,15,1,16 X19,10,20,5

Gauss Code:

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

Jones Polynomial:

q-11 - 2q-10 + 3q-9 - 3q-8 + 4q-7 - 2q-6 + 3q-5 - q-4 + q-3

A2 (sl(3)) Invariant:

q-38 + 2q-28 + 2q-26 + 5q-24 + 4q-22 + 4q-20 + 4q-18 + 2q-16 + 2q-14 + q-10

HOMFLY-PT Polynomial:

a6z-2 + 4a6 + 7a6z2 + 5a6z4 + a6z6 - 2a8z-2 - 3a8 + 2a8z2 + 4a8z4 + a8z6 + a10z-2 - 2a10 - 4a10z2 - a10z4 + a12

Kauffman Polynomial:

a6z-2 - 4a6 + 7a6z2 - 5a6z4 + a6z6 - 2a7z-1 + 4a7z - a7z3 - 3a7z5 + a7z7 + 2a8z-2 - 6a8 + 6a8z2 - a8z4 - 3a8z6 + a8z8 - 2a9z-1 + 11a9z3 - 12a9z5 + 3a9z7 + a10z-2 - 3a10 + 3a10z2 + a10z4 - 3a10z6 + a10z8 - 6a11z + 14a11z3 - 9a11z5 + 2a11z7 - a12 + 5a12z2 - 3a12z4 + a12z6 - 2a13z + 2a13z3 - a14 + a14z2

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -5                 1 j = -7               1 1 j = -9             2     j = -11         1 1 1     j = -13         4 2       j = -15     1 1 1         j = -17     3 3           j = -19   1 2             j = -21   1               j = -23 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]=	3
In[3]:=	Show[DrawMorseLink[Link[10, NonAlternating, 68]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 68]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 12, 6, 13], X[3, 8, 4, 9], X[13, 2, 14, 3], > X[14, 7, 15, 8], X[9, 18, 10, 19], X[11, 16, 12, 17], X[17, 20, 18, 11], > X[4, 15, 1, 16], X[19, 10, 20, 5]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 4, -3, -9}, {-2, -1, 5, 3, -6, 10}, > {-7, 2, -4, -5, 9, 7, -8, 6, -10, 8}]
In[6]:=	Jones[L][q]
Out[6]=	-11 2 3 3 4 2 3 -4 -3 q - --- + -- - -- + -- - -- + -- - q + q 10 9 8 7 6 5 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-38 2 2 5 4 4 4 2 2 -10 q + --- + --- + --- + --- + --- + --- + --- + --- + q 28 26 24 22 20 18 16 14 q q q q q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 68]][a, z]
Out[8]=	6 8 10 6 8 10 12 a 2 a a 6 2 8 2 10 2 4 a - 3 a - 2 a + a + -- - ---- + --- + 7 a z + 2 a z - 4 a z + 2 2 2 z z z 6 4 8 4 10 4 6 6 8 6 > 5 a z + 4 a z - a z + a z + a z
In[9]:=	Kauffman[Link[10, NonAlternating, 68]][a, z]
Out[9]=	6 8 10 7 9 6 8 10 12 14 a 2 a a 2 a 2 a 7 -4 a - 6 a - 3 a - a - a + -- + ---- + --- - ---- - ---- + 4 a z - 2 2 2 z z z z z 11 13 6 2 8 2 10 2 12 2 14 2 > 6 a z - 2 a z + 7 a z + 6 a z + 3 a z + 5 a z + a z - 7 3 9 3 11 3 13 3 6 4 8 4 10 4 > a z + 11 a z + 14 a z + 2 a z - 5 a z - a z + a z - 12 4 7 5 9 5 11 5 6 6 8 6 10 6 > 3 a z - 3 a z - 12 a z - 9 a z + a z - 3 a z - 3 a z + 12 6 7 7 9 7 11 7 8 8 10 8 > a z + a z + 3 a z + 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-7 -5 1 1 1 2 3 1 3 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 23 8 21 7 19 7 19 6 17 6 15 6 17 5 q t q t q t q t q t q t q t 1 1 4 1 2 1 1 2 1 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ---- 15 5 15 4 13 4 11 4 13 3 11 3 11 2 9 2 7 q t q t q t q t q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n68

L10n67 L10n69