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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X18,13,5,14 X14,7,15,8 X8,17,9,18 X16,9,17,10 X10,15,11,16 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

q-21/2 - 2q-19/2 + 4q-17/2 - 6q-15/2 + 7q-13/2 - 8q-11/2 + 6q-9/2 - 6q-7/2 + 3q-5/2 - q-3/2

A2 (sl(3)) Invariant:

- q-34 - 2q-32 - q-28 - q-26 + 3q-24 + q-22 + 2q-20 + 2q-18 + q-16 + 3q-14 + 2q-10 + q-8 - 2q-6 + q-4

HOMFLY-PT Polynomial:

- a3z3 - a5z-1 - 4a5z - 3a5z3 - a7z - 2a7z3 + 2a9z-1 + 3a9z - a11z-1

Kauffman Polynomial:

- a3z3 - 3a4z4 + a5z-1 - 4a5z + 7a5z3 - 6a5z5 - a6 - a6z2 + 7a6z4 - 6a6z6 + 3a7z3 + 3a7z5 - 4a7z7 + 3a8 - 13a8z2 + 21a8z4 - 7a8z6 - a8z8 - 2a9z-1 + 6a9z - 12a9z3 + 16a9z5 - 6a9z7 + 5a10 - 17a10z2 + 15a10z4 - 2a10z6 - a10z8 - a11z-1 + 2a11z - 7a11z3 + 7a11z5 - 2a11z7 + 2a12 - 5a12z2 + 4a12z4 - a12z6

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -2                   1 j = -4                 3 1 j = -6               3     j = -8             3 3     j = -10           5 3       j = -12         3 4         j = -14       3 4           j = -16     1 3             j = -18   1 3               j = -20   1                 j = -22 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[9, Alternating, 13]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 13]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[18, 13, 5, 14], X[14, 7, 15, 8], > X[8, 17, 9, 18], X[16, 9, 17, 10], X[10, 15, 11, 16], X[2, 5, 3, 6], > X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 2, -9}, {8, -1, 4, -5, 6, -7, 9, -2, 3, -4, 7, -6, 5, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 2 4 6 7 8 6 6 3 -(3/2) q - ----- + ----- - ----- + ----- - ----- + ---- - ---- + ---- - q 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 2 -28 -26 3 -22 2 2 -16 3 2 -8 -q - --- - q - q + --- + q + --- + --- + q + --- + --- + q - 32 24 20 18 14 10 q q q q q q 2 -4 > -- + q 6 q
In[8]:=	HOMFLYPT[Link[9, Alternating, 13]][a, z]
Out[8]=	5 9 11 a 2 a a 5 7 9 3 3 5 3 7 3 -(--) + ---- - --- - 4 a z - a z + 3 a z - a z - 3 a z - 2 a z z z z
In[9]:=	Kauffman[Link[9, Alternating, 13]][a, z]
Out[9]=	5 9 11 6 8 10 12 a 2 a a 5 9 11 -a + 3 a + 5 a + 2 a + -- - ---- - --- - 4 a z + 6 a z + 2 a z - z z z 6 2 8 2 10 2 12 2 3 3 5 3 7 3 > a z - 13 a z - 17 a z - 5 a z - a z + 7 a z + 3 a z - 9 3 11 3 4 4 6 4 8 4 10 4 12 4 > 12 a z - 7 a z - 3 a z + 7 a z + 21 a z + 15 a z + 4 a z - 5 5 7 5 9 5 11 5 6 6 8 6 10 6 > 6 a z + 3 a z + 16 a z + 7 a z - 6 a z - 7 a z - 2 a z - 12 6 7 7 9 7 11 7 8 8 10 8 > a z - 4 a z - 6 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 -2 1 1 1 3 1 3 3 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 22 9 20 8 18 8 18 7 16 7 16 6 14 6 q t q t q t q t q t q t q t 4 3 4 5 3 3 3 3 3 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- 14 5 12 5 12 4 10 4 10 3 8 3 8 2 6 2 4 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 L9a13

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