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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X5,14,6,15 X8493 X2,16,3,15 X16,7,17,8 X9,18,10,19 X11,21,12,20 X19,22,20,13 X13,12,14,5 X4,17,1,18 X21,11,22,10

Gauss Code:

{{1, -4, 3, -10}, {-2, -1, 5, -3, -6, 11, -7, 9}, {-9, 2, 4, -5, 10, 6, -8, 7, -11, 8}}

Jones Polynomial:

- q-8 + 3q-7 - 6q-6 + 9q-5 - 10q-4 + 12q-3 - 9q-2 + 8q-1 - 4 + 2q

A2 (sl(3)) Invariant:

- q-24 + q-22 - 2q-20 - q-18 + 3q-16 + 5q-12 + 3q-10 + 4q-8 + 5q-6 + q-4 + 5q-2 + 1 + q2 + 2q4

HOMFLY-PT Polynomial:

z-2 + 3 + 2z2 - 2a2z-2 - 7a2 - 8a2z2 - 3a2z4 + a4z-2 + 6a4 + 8a4z2 + 4a4z4 + a4z6 - 2a6 - 2a6z2 - a6z4

Kauffman Polynomial:

- z-2 + 4 - 6z2 + 3z4 + 2az-1 - 3az - az3 + az5 + az7 - 2a2z-2 + 9a2 - 18a2z2 + 13a2z4 - 4a2z6 + 2a2z8 + 2a3z-1 - 6a3z + 5a3z3 - 3a3z5 + 2a3z7 + a3z9 - a4z-2 + 7a4 - 11a4z2 + 10a4z4 - 8a4z6 + 5a4z8 - 3a5z + 8a5z3 - 11a5z5 + 5a5z7 + a5z9 + 4a6z2 - 6a6z4 - a6z6 + 3a6z8 + a7z - 6a7z5 + 4a7z7 - a8 + 3a8z2 - 6a8z4 + 3a8z6 + a9z - 2a9z3 + a9z5

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 3                   2 j = 1                 2   j = -1               6 2   j = -3             5 4     j = -5           7 4       j = -7         5 7         j = -9       4 5           j = -11     2 5             j = -13   1 4               j = -15   2                 j = -17 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[11, NonAlternating, 325]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 325]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 14, 6, 15], X[8, 4, 9, 3], X[2, 16, 3, 15], > X[16, 7, 17, 8], X[9, 18, 10, 19], X[11, 21, 12, 20], X[19, 22, 20, 13], > X[13, 12, 14, 5], X[4, 17, 1, 18], X[21, 11, 22, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -10}, {-2, -1, 5, -3, -6, 11, -7, 9}, > {-9, 2, 4, -5, 10, 6, -8, 7, -11, 8}]
In[6]:=	Jones[L][q]
Out[6]=	-8 3 6 9 10 12 9 8 -4 - q + -- - -- + -- - -- + -- - -- + - + 2 q 7 6 5 4 3 2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -22 2 -18 3 5 3 4 5 -4 5 2 4 1 - q + q - --- - q + --- + --- + --- + -- + -- + q + -- + q + 2 q 20 16 12 10 8 6 2 q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 325]][a, z]
Out[8]=	2 4 2 4 6 -2 2 a a 2 2 2 4 2 6 2 3 - 7 a + 6 a - 2 a + z - ---- + -- + 2 z - 8 a z + 8 a z - 2 a z - 2 2 z z 2 4 4 4 6 4 4 6 > 3 a z + 4 a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 325]][a, z]
Out[9]=	2 4 3 2 4 8 -2 2 a a 2 a 2 a 3 5 4 + 9 a + 7 a - a - z - ---- - -- + --- + ---- - 3 a z - 6 a z - 3 a z + 2 2 z z z z 7 9 2 2 2 4 2 6 2 8 2 3 > a z + a z - 6 z - 18 a z - 11 a z + 4 a z + 3 a z - a z + 3 3 5 3 9 3 4 2 4 4 4 6 4 > 5 a z + 8 a z - 2 a z + 3 z + 13 a z + 10 a z - 6 a z - 8 4 5 3 5 5 5 7 5 9 5 2 6 4 6 > 6 a z + a z - 3 a z - 11 a z - 6 a z + a z - 4 a z - 8 a z - 6 6 8 6 7 3 7 5 7 7 7 2 8 4 8 > a z + 3 a z + a z + 2 a z + 5 a z + 4 a z + 2 a z + 5 a z + 6 8 3 9 5 9 > 3 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 6 1 2 1 4 2 5 4 5 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 5 7 7 4 5 2 t 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 2 q t + 2 q t 7 3 7 2 5 2 5 3 q q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n325

L11n324 L11n326