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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X14,8,15,7 X15,18,16,5 X11,16,12,17 X17,12,18,13 X8,14,9,13 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-13/2 - 3q-11/2 + 3q-9/2 - 4q-7/2 + 3q-5/2 - 3q-3/2 + 2q-1/2 - q1/2

A2 (sl(3)) Invariant:

- q-20 + q-18 + q-16 + 2q-14 + 3q-12 + q-10 + 2q-8 - q-6 + q2

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- 2az + 3az3 - az5 - 3a2z2 + 6a2z4 - 2a2z6 - a3z-1 + a3z + 2a3z3 + a3z5 - a3z7 + a4 - 4a4z2 + 7a4z4 - 3a4z6 - a5z-1 + 4a5z - 4a5z3 + 2a5z5 - a5z7 - 2a6z2 + a6z4 - a6z6 + a7z - 3a7z3 - a8z2

Khovanov Homology:

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

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L9n11

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