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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X5,12,6,13 X8493 X2,14,3,13 X14,7,15,8 X11,16,12,17 X9,11,10,18 X17,5,18,10 X4,15,1,16

Gauss Code:

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

Jones Polynomial:

- q-5 + 2q-4 - 2q-3 + 3q-2 - 2q-1 + 4 - q + q2

A2 (sl(3)) Invariant:

- q-16 + 2q-8 + 2q-6 + 4q-4 + 5q-2 + 5 + 5q2 + 2q4 + 2q6 + q8

HOMFLY-PT Polynomial:

a-2z-2 + a-2 - 2z-2 - 3 - 2z2 + a2z-2 + 3a2 + 3a2z2 + a2z4 - a4 - a4z2

Kauffman Polynomial:

a-2z-2 - 2a-2 + a-2z2 - 2a-1z-1 + a-1z + a-1z3 + 2z-2 - 5 + 9z2 - 4z4 + z6 - 2az-1 + 3az + 2az3 - 3az5 + az7 + a2z-2 - 6a2 + 13a2z2 - 11a2z4 + 3a2z6 + 3a3z - 2a3z3 - 2a3z5 + a3z7 - 2a4 + 5a4z2 - 7a4z4 + 2a4z6 + a5z - 3a5z3 + a5z5

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 5               1 j = 3                 j = 1           4 1   j = -1         2 4     j = -3       1 1 1     j = -5     1 2         j = -7   1 1           j = -9   1             j = -11 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[9, NonAlternating, 25]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, NonAlternating, 25]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 12, 6, 13], X[8, 4, 9, 3], X[2, 14, 3, 13], > X[14, 7, 15, 8], X[11, 16, 12, 17], X[9, 11, 10, 18], X[17, 5, 18, 10], > X[4, 15, 1, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -9}, {-2, -1, 5, -3, -7, 8}, {-6, 2, 4, -5, 9, 6, -8, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-5 2 2 3 2 2 4 - q + -- - -- + -- - - - q + q 4 3 2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 2 2 4 5 2 4 6 8 5 - q + -- + -- + -- + -- + 5 q + 2 q + 2 q + q 8 6 4 2 q q q q
In[8]:=	HOMFLYPT[Link[9, NonAlternating, 25]][a, z]
Out[8]=	2 -2 2 4 2 1 a 2 2 2 4 2 2 4 -3 + a + 3 a - a - -- + ----- + -- - 2 z + 3 a z - a z + a z 2 2 2 2 z a z z
In[9]:=	Kauffman[Link[9, NonAlternating, 25]][a, z]
Out[9]=	2 2 2 4 2 1 a 2 2 a z 3 -5 - -- - 6 a - 2 a + -- + ----- + -- - --- - --- + - + 3 a z + 3 a z + 2 2 2 2 2 a z z a a z a z z 2 3 5 2 z 2 2 4 2 z 3 3 3 5 3 > a z + 9 z + -- + 13 a z + 5 a z + -- + 2 a z - 2 a z - 3 a z - 2 a a 4 2 4 4 4 5 3 5 5 5 6 2 6 > 4 z - 11 a z - 7 a z - 3 a z - 2 a z + a z + z + 3 a z + 4 6 7 3 7 > 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-3 4 1 1 1 1 1 2 1 1 q + - + 4 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t 2 5 2 > --- + q t + q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L9n25

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