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L11n258 L11n260

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DrawMorseLink

PD Presentation:

X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X11,18,12,19 X22,20,9,19 X20,16,21,15 X16,22,17,21 X17,12,18,13 X2536 X9,1,10,4

Gauss Code:

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

Jones Polynomial:

- q-2 + 4q-1 - 5 + 8q - 7q2 + 8q3 - 6q4 + 4q5 - q6

A2 (sl(3)) Invariant:

- q-6 + 2q-4 + q-2 + 3 + 5q2 + 2q4 + 6q6 + 2q8 + 4q10 + 2q12 + 2q16 - q18

HOMFLY-PT Polynomial:

a-4z-2 + a-4 - a-4z2 - a-4z4 - 2a-2z-2 - 3a-2 + a-2z2 + 3a-2z4 + a-2z6 + z-2 + 2 - z2 - z4

Kauffman Polynomial:

a-7z3 - a-6z2 + 4a-6z4 + 2a-5z5 + a-5z7 - a-4z-2 + 3a-4 - 3a-4z2 - a-4z4 + 2a-4z6 + a-4z8 + 2a-3z-1 - 3a-3z + a-3z3 - 6a-3z5 + 5a-3z7 - 2a-2z-2 + 5a-2 + a-2z2 - 14a-2z4 + 6a-2z6 + a-2z8 + 2a-1z-1 - 3a-1z + a-1z3 - 7a-1z5 + 4a-1z7 - z-2 + 3 + 3z2 - 9z4 + 4z6 - az3 + az5

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 13                 1 j = 11               3   j = 9             4 2   j = 7           4 2     j = 5         3 4       j = 3       5 4         j = 1     3 6           j = -1   1 2             j = -3   3               j = -5 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 259]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 259]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 11, 4, 10], X[7, 15, 8, 14], X[13, 5, 14, 8], > X[11, 18, 12, 19], X[22, 20, 9, 19], X[20, 16, 21, 15], X[16, 22, 17, 21], > X[17, 12, 18, 13], X[2, 5, 3, 6], X[9, 1, 10, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -2, 11}, {10, -1, -3, 4}, > {-11, 2, -5, 9, -4, 3, 7, -8, -9, 5, 6, -7, 8, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-2 4 2 3 4 5 6 -5 - q + - + 8 q - 7 q + 8 q - 6 q + 4 q - q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 2 -2 2 4 6 8 10 12 16 18 3 - q + -- + q + 5 q + 2 q + 6 q + 2 q + 4 q + 2 q + 2 q - q 4 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 259]][a, z]
Out[8]=	2 2 4 4 6 -4 3 -2 1 2 2 z z 4 z 3 z z 2 + a - -- + z + ----- - ----- - z - -- + -- - z - -- + ---- + -- 2 4 2 2 2 4 2 4 2 2 a a z a z a a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 259]][a, z]
Out[9]=	2 2 3 5 -2 1 2 2 2 3 z 3 z 2 z 3 z 3 + -- + -- - z - ----- - ----- + ---- + --- - --- - --- + 3 z - -- - ---- + 4 2 4 2 2 2 3 a z 3 a 6 4 a a a z a z a z a a a 2 3 3 3 4 4 4 5 5 5 z z z z 3 4 4 z z 14 z 2 z 6 z 7 z > -- + -- + -- + -- - a z - 9 z + ---- - -- - ----- + ---- - ---- - ---- + 2 7 3 a 6 4 2 5 3 a a a a a a a a a 6 6 7 7 7 8 8 5 6 2 z 6 z z 5 z 4 z z z > a z + 4 z + ---- + ---- + -- + ---- + ---- + -- + -- 4 2 5 3 a 4 2 a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 3 1 2 3 q 3 5 5 2 6 q + 5 q + ----- + ----- + ---- + --- + --- + 4 q t + 3 q t + 4 q t + 5 3 3 2 2 q t t q t q t q t 7 2 7 3 9 3 9 4 11 4 13 5 > 4 q t + 2 q t + 4 q t + 2 q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n259

L11n258 L11n260