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L10n104 L10n106

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X5,12,6,13 X3849 X2,16,3,15 X7,17,8,16 X14,9,11,10 X20,13,15,14 X19,5,20,10 X11,18,12,19 X17,1,18,4

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

2q-16 + 5q-14 + 6q-12 + 9q-10 + 10q-8 + 10q-6 + 11q-4 + 9q-2 + 9 + 5q2 + 3q4 + 2q6

HOMFLY-PT Polynomial:

- a-1z-3 - 3a-1z-1 - 2a-1z + 3az-3 + 8az-1 + 6az + 2az3 - 3a3z-3 - 7a3z-1 - 4a3z + a5z-3 + 2a5z-1

Kauffman Polynomial:

a-1z-3 - 5a-1z-1 + 8a-1z - 3a-1z3 - 3z-2 + 10 - 10z2 + 4z4 - z6 + 3az-3 - 12az-1 + 19az - 15az3 + 5az5 - az7 - 6a2z-2 + 19a2 - 20a2z2 + 8a2z4 - 2a2z6 + 3a3z-3 - 12a3z-1 + 19a3z - 15a3z3 + 5a3z5 - a3z7 - 3a4z-2 + 10a4 - 10a4z2 + 4a4z4 - a4z6 + a5z-3 - 5a5z-1 + 8a5z - 3a5z3

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 4             2 j = 2           1 2 j = 0         3     j = -2       1 3     j = -4     3 1       j = -6     3         j = -8 2 1           j = -10 2

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, 105]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 105]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 12, 6, 13], X[3, 8, 4, 9], X[2, 16, 3, 15], > X[7, 17, 8, 16], X[14, 9, 11, 10], X[20, 13, 15, 14], X[19, 5, 20, 10], > X[11, 18, 12, 19], X[17, 1, 18, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, -3, 10}, {-9, 2, 7, -6}, {-2, -1, -5, 3, 6, 8}, > {4, 5, -10, 9, -8, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-2 -(7/2) 4 2 4 3/2 ---- + q - ---- + ---- - ------- + Sqrt[q] - 2 q 9/2 5/2 3/2 Sqrt[q] q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 5 6 9 10 10 11 9 2 4 6 9 + --- + --- + --- + --- + -- + -- + -- + -- + 5 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, 105]][a, z]
Out[8]=	3 5 3 5 1 3 a 3 a a 3 8 a 7 a 2 a 2 z 3 -(----) + --- - ---- + -- - --- + --- - ---- + ---- - --- + 6 a z - 4 a z + 3 3 3 3 a z z z z a a z z z z 3 > 2 a z
In[9]:=	Kauffman[Link[10, NonAlternating, 105]][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 8 z 3 5 2 2 2 > ----- - ---- + --- + 19 a z + 19 a z + 8 a z - 10 z - 20 a z - z z a 3 4 2 3 z 3 3 3 5 3 4 2 4 4 4 > 10 a z - ---- - 15 a z - 15 a z - 3 a z + 4 z + 8 a z + 4 a z + a 5 3 5 6 2 6 4 6 7 3 7 > 5 a z + 5 a z - z - 2 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 2 2 1 3 3 1 1 2 3 + -- + ------ + ----- + ----- + ----- + ----- + ---- + ---- + q t + 2 10 4 8 4 8 3 6 2 4 2 4 2 q q t q t q t q t q t q t q t 2 2 4 2 > 2 q t + 2 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n105

L10n104 L10n106