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L11n243 L11n245

Knotscape

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DrawMorseLink

PD Presentation:

X12,1,13,2 X10,11,1,12 X5,15,6,14 X9,19,10,18 X17,3,18,2 X7,16,8,17 X3849 X15,20,16,21 X22,13,11,14 X19,4,20,5 X21,7,22,6

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:

- q-13/2 + 2q-11/2 - q-9/2 + q-7/2 - q-5/2 - q-3/2 - 2q1/2 + 2q3/2 - q5/2

A2 (sl(3)) Invariant:

q-20 - 2q-14 - 2q-12 + q-8 + 4q-6 + 2q-4 + 2q-2 + 1 + q4 + q8

HOMFLY-PT Polynomial:

- a-1z-1 - 2a-1z - a-1z3 + az-1 + 6az + 5az3 + az5 - 6a3z - 5a3z3 - a3z5 + 2a5z + a5z3

Kauffman Polynomial:

- a-1z-1 + 4a-1z - 6a-1z3 + 5a-1z5 - a-1z7 + 1 + 4z2 - 14z4 + 11z6 - 2z8 - az-1 + 12az - 19az3 + 4az5 + 4az7 - az9 + 4a2z2 - 24a2z4 + 18a2z6 - 3a2z8 + 12a3z - 21a3z3 + 5a3z5 + 4a3z7 - a3z9 - 4a4z2 - 2a4z4 + 5a4z6 - a4z8 + 4a5z - 5a5z3 + 5a5z5 - a5z7 - 4a6z2 + 8a6z4 - 2a6z6 + 3a7z3 - a7z5

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 6                     1 j = 4                   1   j = 2                 1 1   j = 0             3 2 1     j = -2           1 3 1       j = -4         2 2 2         j = -6       1 2 1           j = -8     1 2 1             j = -10   1 1 1               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[11, NonAlternating, 244]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 244]]
Out[4]=	PD[X[12, 1, 13, 2], X[10, 11, 1, 12], X[5, 15, 6, 14], X[9, 19, 10, 18], > X[17, 3, 18, 2], X[7, 16, 8, 17], X[3, 8, 4, 9], X[15, 20, 16, 21], > X[22, 13, 11, 14], X[19, 4, 20, 5], X[21, 7, 22, 6]]
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]=	-(13/2) 2 -(9/2) -(7/2) -(5/2) -(3/2) -q + ----- - q + q - q - q - 2 Sqrt[q] + 11/2 q 3/2 5/2 > 2 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 2 2 -8 4 2 2 4 8 1 + q - --- - --- + q + -- + -- + -- + q + q 14 12 6 4 2 q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 244]][a, z]
Out[8]=	3 1 a 2 z 3 5 z 3 3 3 5 3 -(---) + - - --- + 6 a z - 6 a z + 2 a z - -- + 5 a z - 5 a z + a z + a z z a a 5 3 5 > a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 244]][a, z]
Out[9]=	1 a 4 z 3 5 2 2 2 4 2 1 - --- - - + --- + 12 a z + 12 a z + 4 a z + 4 z + 4 a z - 4 a z - a z z a 3 6 2 6 z 3 3 3 5 3 7 3 4 > 4 a z - ---- - 19 a z - 21 a z - 5 a z + 3 a z - 14 z - a 5 2 4 4 4 6 4 5 z 5 3 5 5 5 7 5 > 24 a z - 2 a z + 8 a z + ---- + 4 a z + 5 a z + 5 a z - a z + a 7 6 2 6 4 6 6 6 z 7 3 7 5 7 > 11 z + 18 a z + 5 a z - 2 a z - -- + 4 a z + 4 a z - a z - a 8 2 8 4 8 9 3 9 > 2 z - 3 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 3 1 1 1 1 1 1 2 3 + -- + -- + ------ + ------ + ------ + ------ + ----- + ------ + ----- + 4 2 14 6 12 5 10 5 10 4 8 4 10 3 8 3 q q q t q t q t q t q t q t q t 1 1 2 2 1 2 1 t 2 > ----- + ----- + ----- + ----- + ---- + ---- + ---- + 2 t + -- + t + 6 3 8 2 6 2 4 2 6 4 2 2 q t q t q t q t q t q t q t q 2 2 2 3 4 3 6 4 > q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n244

L11n243 L11n245