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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X4,15,1,16 X5,10,6,11 X3849 X11,19,12,18 X17,5,18,20 X19,13,20,12 X9,16,10,17 X13,2,14,3

Gauss Code:

{{1, 10, -5, -3}, {-4, -1, 2, 5, -9, 4, -6, 8, -10, -2, 3, 9, -7, 6, -8, 7}}

Jones Polynomial:

q-13/2 - 2q-11/2 + 2q-9/2 - 4q-7/2 + 3q-5/2 - 3q-3/2 + 2q-1/2 - 2q1/2 + q3/2

A2 (sl(3)) Invariant:

- q-24 - q-20 + q-16 + 2q-14 + 4q-12 + 2q-10 + 3q-8 - q4

HOMFLY-PT Polynomial:

az-1 + 3az + 4az3 + az5 - 4a3z-1 - 11a3z - 12a3z3 - 6a3z5 - a3z7 + 4a5z-1 + 7a5z + 5a5z3 + a5z5 - a7z-1 - a7z

Kauffman Polynomial:

1 - 3z2 + 4z4 - z6 - az-1 + 4az - 10az3 + 9az5 - 2az7 + 4a2 - 11a2z2 + 6a2z4 + 2a2z6 - a2z8 - 4a3z-1 + 16a3z - 27a3z3 + 19a3z5 - 4a3z7 + 7a4 - 15a4z2 + 6a4z4 + 2a4z6 - a4z8 - 4a5z-1 + 15a5z - 19a5z3 + 10a5z5 - 2a5z7 + 4a6 - 8a6z2 + 4a6z4 - a6z6 - a7z-1 + 3a7z - 2a7z3 + a8 - a8z2

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 4                 1 j = 2               1   j = 0             1 1   j = -2         1 3 1     j = -4         2 2       j = -6       3 2         j = -8   1 1 2           j = -10   2 2             j = -12   1               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[10, NonAlternating, 1]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 1]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[5, 10, 6, 11], > X[3, 8, 4, 9], X[11, 19, 12, 18], X[17, 5, 18, 20], X[19, 13, 20, 12], > X[9, 16, 10, 17], X[13, 2, 14, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 10, -5, -3}, {-4, -1, 2, 5, -9, 4, -6, 8, -10, -2, 3, 9, -7, 6, > -8, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 2 2 4 3 3 2 3/2 q - ----- + ---- - ---- + ---- - ---- + ------- - 2 Sqrt[q] + 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]=	-24 -20 -16 2 4 2 3 4 -q - q + q + --- + --- + --- + -- - q 14 12 10 8 q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 1]][a, z]
Out[8]=	3 5 7 a 4 a 4 a a 3 5 7 3 3 3 - - ---- + ---- - -- + 3 a z - 11 a z + 7 a z - a z + 4 a z - 12 a z + z z z z 5 3 5 3 5 5 5 3 7 > 5 a z + a z - 6 a z + a z - a z
In[9]:=	Kauffman[Link[10, NonAlternating, 1]][a, z]
Out[9]=	3 5 7 2 4 6 8 a 4 a 4 a a 3 1 + 4 a + 7 a + 4 a + a - - - ---- - ---- - -- + 4 a z + 16 a z + z z z z 5 7 2 2 2 4 2 6 2 8 2 3 > 15 a z + 3 a z - 3 z - 11 a z - 15 a z - 8 a z - a z - 10 a z - 3 3 5 3 7 3 4 2 4 4 4 6 4 > 27 a z - 19 a z - 2 a z + 4 z + 6 a z + 6 a z + 4 a z + 5 3 5 5 5 6 2 6 4 6 6 6 7 > 9 a z + 19 a z + 10 a z - z + 2 a z + 2 a z - a z - 2 a z - 3 7 5 7 2 8 4 8 > 4 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 3 1 1 2 1 2 1 2 3 -- + -- + ------ + ------ + ------ + ----- + ------ + ----- + ----- + ----- + 4 2 14 5 12 4 10 4 8 4 10 3 8 3 8 2 6 2 q q q t q t q t q t q t q t q t q t 2 2 1 t 2 2 2 4 3 > ---- + ---- + ---- + t + -- + t + q t + q t 6 4 2 2 q t q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n1

L10a174 L10n2