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L11n297 L11n299

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DrawMorseLink

PD Presentation:

X6172 X3,13,4,12 X13,19,14,18 X17,11,18,22 X7,17,8,16 X21,8,22,9 X9,14,10,15 X15,20,16,21 X19,5,20,10 X2536 X11,1,12,4

Gauss Code:

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

Jones Polynomial:

- 2q-3 + 7q-2 - 9q-1 + 14 - 14q + 14q2 - 11q3 + 8q4 - 4q5 + q6

A2 (sl(3)) Invariant:

- 2q-10 + 2q-8 + 4q-6 + q-4 + 8q-2 + 4 + 5q2 + 4q4 - q6 + 3q8 - 3q10 + q12 + 2q14 - 2q16 + q18

HOMFLY-PT Polynomial:

a-4 + a-4z2 + a-4z4 + a-2z-2 - 2a-2 - 5a-2z2 - 3a-2z4 - a-2z6 - 2z-2 + 1 + 5z2 + 3z4 + a2z-2 - 2a2z2

Kauffman Polynomial:

a-6z2 - 2a-6z4 + a-6z6 - 2a-5z + 6a-5z3 - 10a-5z5 + 4a-5z7 + 2a-4 - 3a-4z2 + 8a-4z4 - 15a-4z6 + 6a-4z8 - 7a-3z + 25a-3z3 - 27a-3z5 + 3a-3z7 + 3a-3z9 + a-2z-2 + 4a-2 - 20a-2z2 + 39a-2z4 - 40a-2z6 + 14a-2z8 - 2a-1z-1 - 7a-1z + 30a-1z3 - 28a-1z5 + 5a-1z7 + 3a-1z9 + 2z-2 + 3 - 25z2 + 37z4 - 23z6 + 8z8 - 2az-1 - 3az + 14az3 - 11az5 + 6az7 + a2z-2 - 9a2z2 + 8a2z4 + a2z6 - a3z + 3a3z3

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                   1 j = 11                 3   j = 9               5 1   j = 7             6 3     j = 5           8 5       j = 3         7 7         j = 1       7 7           j = -1     4 9             j = -3   3 5               j = -5   5                 j = -7 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 298]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 298]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 13, 4, 12], X[13, 19, 14, 18], X[17, 11, 18, 22], > X[7, 17, 8, 16], X[21, 8, 22, 9], X[9, 14, 10, 15], X[15, 20, 16, 21], > X[19, 5, 20, 10], X[2, 5, 3, 6], X[11, 1, 12, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -2, 11}, {10, -1, -5, 6, -7, 9}, > {-11, 2, -3, 7, -8, 5, -4, 3, -9, 8, -6, 4}]
In[6]:=	Jones[L][q]
Out[6]=	2 7 9 2 3 4 5 6 14 - -- + -- - - - 14 q + 14 q - 11 q + 8 q - 4 q + q 3 2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 2 4 -4 8 2 4 6 8 10 12 14 4 - --- + -- + -- + q + -- + 5 q + 4 q - q + 3 q - 3 q + q + 2 q - 10 8 6 2 q q q q 16 18 > 2 q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 298]][a, z]
Out[8]=	2 2 2 4 -4 2 2 1 a 2 z 5 z 2 2 4 z 1 + a - -- - -- + ----- + -- + 5 z + -- - ---- - 2 a z + 3 z + -- - 2 2 2 2 2 4 2 4 a z a z z a a a 4 6 3 z z > ---- - -- 2 2 a a
In[9]:=	Kauffman[Link[11, NonAlternating, 298]][a, z]
Out[9]=	2 2 4 2 1 a 2 2 a 2 z 7 z 7 z 3 3 + -- + -- + -- + ----- + -- - --- - --- - --- - --- - --- - 3 a z - a z - 4 2 2 2 2 2 a z z 5 3 a a a z a z z a a 2 2 2 3 3 3 2 z 3 z 20 z 2 2 6 z 25 z 30 z 3 > 25 z + -- - ---- - ----- - 9 a z + ---- + ----- + ----- + 14 a z + 6 4 2 5 3 a a a a a a 4 4 4 5 5 5 3 3 4 2 z 8 z 39 z 2 4 10 z 27 z 28 z > 3 a z + 37 z - ---- + ---- + ----- + 8 a z - ----- - ----- - ----- - 6 4 2 5 3 a a a a a a 6 6 6 7 7 7 5 6 z 15 z 40 z 2 6 4 z 3 z 5 z > 11 a z - 23 z + -- - ----- - ----- + a z + ---- + ---- + ---- + 6 4 2 5 3 a a a a a a 8 8 9 9 7 8 6 z 14 z 3 z 3 z > 6 a z + 8 z + ---- + ----- + ---- + ---- 4 2 3 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	9 2 5 3 5 4 3 3 2 - + 7 q + ----- + ----- + ----- + ---- + --- + 7 q t + 7 q t + 7 q t + q 7 3 5 2 3 2 3 q t q t q t q t q t 5 2 5 3 7 3 7 4 9 4 9 5 11 5 13 6 > 8 q t + 5 q t + 6 q t + 3 q t + 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n298

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