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DrawMorseLink

PD Presentation:

X6172 X5,12,6,13 X3849 X2,16,3,15 X16,7,17,8 X19,11,20,14 X13,15,14,20 X9,18,10,19 X11,10,12,5 X17,1,18,4

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 2q-11/2 - 4q-9/2 + 4q-7/2 - 7q-5/2 + 4q-3/2 - 6q-1/2 + 2q1/2 - 2q3/2

A2 (sl(3)) Invariant:

q-20 + 2q-16 + 4q-14 + 4q-12 + 9q-10 + 10q-8 + 12q-6 + 12q-4 + 9q-2 + 9 + 4q2 + 3q4 + 2q6

HOMFLY-PT Polynomial:

- a-1z-3 - 3a-1z-1 - 2a-1z + 3az-3 + 8az-1 + 8az + 3az3 - 3a3z-3 - 7a3z-1 - 8a3z - 4a3z3 - a3z5 + a5z-3 + 2a5z-1 + 2a5z + a5z3

Kauffman Polynomial:

a-1z-3 - 5a-1z-1 + 7a-1z - 3a-1z3 - 3z-2 + 10 - 7z2 + 2z4 - z6 + 3az-3 - 12az-1 + 23az - 21az3 + 8az5 - 2az7 - 6a2z-2 + 19a2 - 23a2z2 + 8a2z4 - a2z8 + 3a3z-3 - 12a3z-1 + 29a3z - 36a3z3 + 20a3z5 - 5a3z7 - 3a4z-2 + 10a4 - 17a4z2 + 11a4z4 - a4z6 - a4z8 + a5z-3 - 5a5z-1 + 12a5z - 15a5z3 + 11a5z5 - 3a5z7 - a6z2 + 5a6z4 - 2a6z6 - a7z + 3a7z3 - a7z5

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 4                 2 j = 2               1 1 j = 0             5 1   j = -2           3 5     j = -4         4 1       j = -6       2 5         j = -8     2 2           j = -10     2             j = -12 1 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]=	4
In[3]:=	Show[DrawMorseLink[Link[10, NonAlternating, 109]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 109]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 12, 6, 13], X[3, 8, 4, 9], X[2, 16, 3, 15], > X[16, 7, 17, 8], X[19, 11, 20, 14], X[13, 15, 14, 20], X[9, 18, 10, 19], > X[11, 10, 12, 5], X[17, 1, 18, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, -3, 10}, {-9, 2, -7, 6}, {-2, -1, 5, 3, -8, 9}, > {4, -5, -10, 8, -6, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 2 4 4 7 4 6 3/2 -q + ----- - ---- + ---- - ---- + ---- - ------- + 2 Sqrt[q] - 2 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 2 4 4 9 10 12 12 9 2 4 6 9 + q + --- + --- + --- + --- + -- + -- + -- + -- + 4 q + 3 q + 2 q 16 14 12 10 8 6 4 2 q q q q q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 109]][a, z]
Out[8]=	3 5 3 5 1 3 a 3 a a 3 8 a 7 a 2 a 2 z 3 -(----) + --- - ---- + -- - --- + --- - ---- + ---- - --- + 8 a z - 8 a z + 3 3 3 3 a z z z z a a z z z z 5 3 3 3 5 3 3 5 > 2 a z + 3 a z - 4 a z + a z - a z
In[9]:=	Kauffman[Link[10, NonAlternating, 109]][a, z]
Out[9]=	3 5 2 4 2 4 1 3 a 3 a a 3 6 a 3 a 5 12 a 10 + 19 a + 10 a + ---- + --- + ---- + -- - -- - ---- - ---- - --- - ---- - 3 3 3 3 2 2 2 a z z a z z z z z z z 3 5 12 a 5 a 7 z 3 5 7 2 2 2 > ----- - ---- + --- + 23 a z + 29 a z + 12 a z - a z - 7 z - 23 a z - z z a 3 4 2 6 2 3 z 3 3 3 5 3 7 3 4 > 17 a z - a z - ---- - 21 a z - 36 a z - 15 a z + 3 a z + 2 z + a 2 4 4 4 6 4 5 3 5 5 5 7 5 6 > 8 a z + 11 a z + 5 a z + 8 a z + 20 a z + 11 a z - a z - z - 4 6 6 6 7 3 7 5 7 2 8 4 8 > a z - 2 a z - 2 a z - 5 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 1 1 2 2 2 2 2 5 5 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 6 12 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 4 1 3 2 2 2 4 2 > ----- + ---- + ---- + t + q t + q t + 2 q t 4 2 4 2 q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n109

L10n108 L10n110