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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 X16,9,17,10 X8,15,9,16 X10,17,11,18 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

- q-36 - 2q-34 - q-32 - 2q-30 + q-26 + q-24 + 4q-22 + 2q-20 + 4q-18 + 2q-16 + q-12 - q-10 + q-8

HOMFLY-PT Polynomial:

- a5z - 3a5z3 - a5z5 - 3a7z-1 - 9a7z - 8a7z3 - 2a7z5 + 5a9z-1 + 9a9z + 3a9z3 - 2a11z-1 - a11z

Kauffman Polynomial:

- a5z + 3a5z3 - a5z5 - a6z2 + 5a6z4 - 2a6z6 - 3a7z-1 + 10a7z - 15a7z3 + 11a7z5 - 3a7z7 + 5a8 - 12a8z2 + 9a8z4 - a8z6 - a8z8 - 5a9z-1 + 16a9z - 26a9z3 + 17a9z5 - 5a9z7 + 5a10 - 11a10z2 + 6a10z4 - a10z6 - a10z8 - 2a11z-1 + 5a11z - 5a11z3 + 3a11z5 - 2a11z7 + 2a12z2 + a12z4 - 2a12z6 + 3a13z3 - 2a13z5 - a14 + 2a14z2 - a14z4

Khovanov Homology:

trqj 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                 2 1 j = -8               2     j = -10             2 2     j = -12           5 2       j = -14         2 3         j = -16       3 4           j = -18     1 2             j = -20   1 3               j = -22   1                 j = -24 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, 12]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 12]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[18, 13, 5, 14], X[14, 7, 15, 8], > X[16, 9, 17, 10], X[8, 15, 9, 16], X[10, 17, 11, 18], X[2, 5, 3, 6], > X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 2, -9}, {8, -1, 4, -6, 5, -7, 9, -2, 3, -4, 6, -5, 7, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(23/2) 2 4 5 6 7 4 4 2 -(5/2) q - ----- + ----- - ----- + ----- - ----- + ----- - ---- + ---- - q 21/2 19/2 17/2 15/2 13/2 11/2 9/2 7/2 q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-36 2 -32 2 -26 -24 4 2 4 2 -12 -10 -q - --- - q - --- + q + q + --- + --- + --- + --- + q - q + 34 30 22 20 18 16 q q q q q q -8 > q
In[8]:=	HOMFLYPT[Link[9, Alternating, 12]][a, z]
Out[8]=	7 9 11 -3 a 5 a 2 a 5 7 9 11 5 3 7 3 ----- + ---- - ----- - a z - 9 a z + 9 a z - a z - 3 a z - 8 a z + z z z 9 3 5 5 7 5 > 3 a z - a z - 2 a z
In[9]:=	Kauffman[Link[9, Alternating, 12]][a, z]
Out[9]=	7 9 11 8 10 14 3 a 5 a 2 a 5 7 9 11 5 a + 5 a - a - ---- - ---- - ----- - a z + 10 a z + 16 a z + 5 a z - z z z 6 2 8 2 10 2 12 2 14 2 5 3 7 3 > a z - 12 a z - 11 a z + 2 a z + 2 a z + 3 a z - 15 a z - 9 3 11 3 13 3 6 4 8 4 10 4 12 4 > 26 a z - 5 a z + 3 a z + 5 a z + 9 a z + 6 a z + a z - 14 4 5 5 7 5 9 5 11 5 13 5 6 6 > a z - a z + 11 a z + 17 a z + 3 a z - 2 a z - 2 a z - 8 6 10 6 12 6 7 7 9 7 11 7 8 8 10 8 > a z - a z - 2 a z - 3 a z - 5 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 1 1 3 1 2 3 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 24 9 22 8 20 8 20 7 18 7 18 6 16 6 q t q t q t q t q t q t q t 4 2 3 5 2 2 2 2 2 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ---- 16 5 14 5 14 4 12 4 12 3 10 3 10 2 8 2 6 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 L9a12

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