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L11n143 L11n145

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DrawMorseLink

PD Presentation:

X8192 X11,19,12,18 X3,10,4,11 X2,17,3,18 X12,5,13,6 X6718 X16,10,17,9 X20,16,21,15 X22,14,7,13 X14,22,15,21 X19,4,20,5

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 2q-11/2 - 4q-9/2 + 5q-7/2 - 7q-5/2 + 6q-3/2 - 6q-1/2 + 4q1/2 - 2q3/2 + q5/2

A2 (sl(3)) Invariant:

q-20 + q-16 + 2q-14 + 3q-10 + 2q-8 + 2q-6 + 2q-4 - q-2 + 1 - 2q2 - q4 - q8

HOMFLY-PT Polynomial:

a-1z-1 + 2a-1z + a-1z3 - 2az-1 - 4az - 3az3 - az5 - 3a3z - 3a3z3 - a3z5 + a5z-1 + 2a5z + a5z3

Kauffman Polynomial:

- 2a-2 + 3a-2z2 - a-2z4 + a-1z-1 - 2a-1z + 4a-1z3 - 2a-1z5 - 5 + 17z2 - 15z4 + 5z6 - z8 + 2az-1 - 7az + 14az3 - 14az5 + 5az7 - az9 - 3a2 + 13a2z2 - 22a2z4 + 12a2z6 - 3a2z8 + 2a3z - 4a3z3 - a3z5 + 2a3z7 - a3z9 + a4 - 2a4z2 - 3a4z4 + 5a4z6 - 2a4z8 - a5z-1 + 6a5z - 11a5z3 + 10a5z5 - 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 r = 3 j = 6                   1 j = 4                 1   j = 2               3 1   j = 0             3 1     j = -2           4 4       j = -4         3 2         j = -6       2 4           j = -8     2 3             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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 144]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 144]]
Out[4]=	PD[X[8, 1, 9, 2], X[11, 19, 12, 18], X[3, 10, 4, 11], X[2, 17, 3, 18], > X[12, 5, 13, 6], X[6, 7, 1, 8], X[16, 10, 17, 9], X[20, 16, 21, 15], > X[22, 14, 7, 13], X[14, 22, 15, 21], X[19, 4, 20, 5]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, -3, 11, 5, -6}, > {6, -1, 7, 3, -2, -5, 9, -10, 8, -7, 4, 2, -11, -8, 10, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 2 4 5 7 6 6 3/2 -q + ----- - ---- + ---- - ---- + ---- - ------- + 4 Sqrt[q] - 2 q + 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 5/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -16 2 3 2 2 2 -2 2 4 8 1 + q + q + --- + --- + -- + -- + -- - q - 2 q - q - q 14 10 8 6 4 q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 144]][a, z]
Out[8]=	5 3 1 2 a a 2 z 3 5 z 3 3 3 --- - --- + -- + --- - 4 a z - 3 a z + 2 a z + -- - 3 a z - 3 a z + a z z z a a 5 3 5 3 5 > a z - a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 144]][a, z]
Out[9]=	5 2 2 4 1 2 a a 2 z 3 5 7 -5 - -- - 3 a + a + --- + --- - -- - --- - 7 a z + 2 a z + 6 a z - a z + 2 a z z z a a 2 3 2 3 z 2 2 4 2 6 2 4 z 3 3 3 > 17 z + ---- + 13 a z - 2 a z - a z + ---- + 14 a z - 4 a z - 2 a a 4 5 5 3 7 3 4 z 2 4 4 4 6 4 2 z > 11 a z + 3 a z - 15 z - -- - 22 a z - 3 a z + 5 a z - ---- - 2 a a 5 3 5 5 5 7 5 6 2 6 4 6 6 6 > 14 a z - a z + 10 a z - a z + 5 z + 12 a z + 5 a z - 2 a z + 7 3 7 5 7 8 2 8 4 8 9 3 9 > 5 a z + 2 a z - 3 a z - z - 3 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 1 1 2 2 2 3 2 4 3 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 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 3 2 4 2 2 2 4 2 6 3 > ----- + ---- + ---- + t + 3 q t + q t + q t + q t 4 2 4 2 q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n144

L11n143 L11n145