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L11n244 L11n246

Knotscape

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DrawMorseLink

PD Presentation:

X12,1,13,2 X10,11,1,12 X14,5,15,6 X9,19,10,18 X17,3,18,2 X16,8,17,7 X3849 X20,16,21,15 X22,13,11,14 X4,20,5,19 X6,21,7,22

Gauss Code:

{{1, 5, -7, -10, 3, -11, 6, 7, -4, -2}, {2, -1, 9, -3, 8, -6, -5, 4, 10, -8, 11, -9}}

Jones Polynomial:

- 2q-9/2 + 6q-7/2 - 9q-5/2 + 11q-3/2 - 13q-1/2 + 11q1/2 - 10q3/2 + 6q5/2 - 3q7/2 + q9/2

A2 (sl(3)) Invariant:

2q-14 - 2q-12 + q-8 - 3q-6 + 3q-4 + 4 + 3q2 + 3q6 - 2q8 + q12 - q14

HOMFLY-PT Polynomial:

a-3z + a-3z3 - a-1z-1 - a-1z3 - a-1z5 + az-1 - 3az - 5az3 - 2az5 + 2a3z + 2a3z3

Kauffman Polynomial:

- 2a-4z2 + 3a-4z4 - a-4z6 + 3a-3z - 8a-3z3 + 9a-3z5 - 3a-3z7 - 5a-2z4 + 10a-2z6 - 4a-2z8 - a-1z-1 + 4a-1z - 13a-1z3 + 16a-1z5 - 2a-1z7 - 2a-1z9 + 1 + 5z2 - 16z4 + 22z6 - 9z8 - az-1 - 4az3 + 10az5 - 3az7 - 2az9 + 8a2z2 - 14a2z4 + 10a2z6 - 5a2z8 - 2a3z3 + 3a3z5 - 4a3z7 + 5a4z2 - 6a4z4 - a4z6 + a5z - 3a5z3

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 10                   1 j = 8                 2   j = 6               4 1   j = 4             6 2     j = 2           5 4       j = 0         8 6         j = -2       5 7           j = -4     4 6             j = -6   2 5               j = -8   4                 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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 245]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 245]]
Out[4]=	PD[X[12, 1, 13, 2], X[10, 11, 1, 12], X[14, 5, 15, 6], X[9, 19, 10, 18], > X[17, 3, 18, 2], X[16, 8, 17, 7], X[3, 8, 4, 9], X[20, 16, 21, 15], > X[22, 13, 11, 14], X[4, 20, 5, 19], X[6, 21, 7, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 5, -7, -10, 3, -11, 6, 7, -4, -2}, > {2, -1, 9, -3, 8, -6, -5, 4, 10, -8, 11, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-2 6 9 11 13 3/2 5/2 7/2 ---- + ---- - ---- + ---- - ------- + 11 Sqrt[q] - 10 q + 6 q - 3 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 9/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 2 -8 3 3 2 6 8 12 14 4 + --- - --- + q - -- + -- + 3 q + 3 q - 2 q + q - q 14 12 6 4 q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 245]][a, z]
Out[8]=	3 3 5 1 a z 3 z z 3 3 3 z 5 -(---) + - + -- - 3 a z + 2 a z + -- - -- - 5 a z + 2 a z - -- - 2 a z a z z 3 3 a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 245]][a, z]
Out[9]=	2 3 1 a 3 z 4 z 5 2 2 z 2 2 4 2 8 z 1 - --- - - + --- + --- + a z + 5 z - ---- + 8 a z + 5 a z - ---- - a z z 3 a 4 3 a a a 3 4 4 13 z 3 3 3 5 3 4 3 z 5 z 2 4 > ----- - 4 a z - 2 a z - 3 a z - 16 z + ---- - ---- - 14 a z - a 4 2 a a 5 5 6 6 4 4 9 z 16 z 5 3 5 6 z 10 z > 6 a z + ---- + ----- + 10 a z + 3 a z + 22 z - -- + ----- + 3 a 4 2 a a a 7 7 8 2 6 4 6 3 z 2 z 7 3 7 8 4 z 2 8 > 10 a z - a z - ---- - ---- - 3 a z - 4 a z - 9 z - ---- - 5 a z - 3 a 2 a a 9 2 z 9 > ---- - 2 a z a
In[10]:=	Kh[L][q, t]
Out[10]=	7 2 4 2 5 4 6 5 2 8 + -- + ------ + ----- + ----- + ----- + ----- + ---- + ---- + 6 t + 5 q t + 2 10 4 8 3 6 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 4 3 6 3 6 4 8 4 10 5 > 4 q t + 6 q t + 2 q t + 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n245

L11n244 L11n246