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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-17/2 - 2q-15/2 + 4q-13/2 - 7q-11/2 + 7q-9/2 - 8q-7/2 + 6q-5/2 - 5q-3/2 + 3q-1/2 - q1/2

A2 (sl(3)) Invariant:

- q-28 - 2q-26 + 4q-18 + 2q-16 + 3q-14 + 2q-12 + 2q-8 - 2q-6 + q-4 - 1 + q2

HOMFLY-PT Polynomial:

- az - az3 + a3z + 2a3z3 + a3z5 - 2a5z-1 - 5a5z - 3a5z3 + 3a7z-1 + 3a7z - a9z-1

Kauffman Polynomial:

- az + 2az3 - az5 - 3a2z2 + 7a2z4 - 3a2z6 + 5a3z5 - 3a3z7 - 5a4z2 + 11a4z4 - 4a4z6 - a4z8 + 2a5z-1 - 4a5z + 7a5z5 - 5a5z7 - 3a6 + a6z2 + 4a6z4 - 3a6z6 - a6z8 + 3a7z-1 - 7a7z + 5a7z3 - a7z5 - 2a7z7 - 3a8 + 5a8z2 - a8z4 - 2a8z6 + a9z-1 - 2a9z + 3a9z3 - 2a9z5 - a10 + 2a10z2 - a10z4

Khovanov Homology:

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

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L9a16

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