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L11n308 L11n310

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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X7,17,8,16 X9,21,10,20 X15,9,16,8 X19,5,20,10 X18,13,19,14 X22,17,11,18 X21,15,22,14 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

q-6 - 3q-5 + 6q-4 - 8q-3 + 10q-2 - 10q-1 + 10 - 6q + 5q2 - q3

A2 (sl(3)) Invariant:

q-20 + q-18 - 2q-16 + q-14 - 2q-10 + 3q-8 + q-6 + 4q-4 + 4q-2 + 3 + 6q2 + q4 + 4q6 + 3q8 - q10

HOMFLY-PT Polynomial:

a-2z-2 + 2a-2 - a-2z2 - 2z-2 - 4 - 2z2 + z4 + a2z-2 + 3a2 + 3a2z2 + 2a2z4 - 2a4 - 3a4z2 + a6

Kauffman Polynomial:

a-3z3 + a-2z-2 - 4a-2 - 2a-2z2 + 5a-2z4 - 2a-1z-1 + 4a-1z + a-1z3 - a-1z5 + 2a-1z7 + 2z-2 - 6 + z2 + 8z4 - 6z6 + 3z8 - 2az-1 + 13az3 - 16az5 + 5az7 + az9 + a2z-2 - 3a2 + 6a2z2 + a2z4 - 12a2z6 + 6a2z8 - 6a3z + 20a3z3 - 24a3z5 + 6a3z7 + a3z9 - a4 + 6a4z2 - 5a4z4 - 5a4z6 + 3a4z8 - 2a5z + 7a5z3 - 9a5z5 + 3a5z7 - a6 + 3a6z2 - 3a6z4 + a6z6

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 7                   1 j = 5                 4   j = 3               4 3   j = 1             6 2     j = -1           5 5       j = -3         5 5         j = -5       3 5           j = -7     3 5             j = -9   1 4               j = -11   2                 j = -13 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, 309]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 309]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[7, 17, 8, 16], X[9, 21, 10, 20], > X[15, 9, 16, 8], X[19, 5, 20, 10], X[18, 13, 19, 14], X[22, 17, 11, 18], > X[21, 15, 22, 14], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -3, 5, -4, 6}, > {11, -2, 7, 9, -5, 3, 8, -7, -6, 4, -9, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-6 3 6 8 10 10 2 3 10 + q - -- + -- - -- + -- - -- - 6 q + 5 q - q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -18 2 -14 2 3 -6 4 4 2 4 6 3 + q + q - --- + q - --- + -- + q + -- + -- + 6 q + q + 4 q + 16 10 8 4 2 q q q q q 8 10 > 3 q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 309]][a, z]
Out[8]=	2 2 2 2 4 6 2 1 a 2 z 2 2 4 2 -4 + -- + 3 a - 2 a + a - -- + ----- + -- - 2 z - -- + 3 a z - 3 a z + 2 2 2 2 2 2 a z a z z a 4 2 4 > z + 2 a z
In[9]:=	Kauffman[Link[11, NonAlternating, 309]][a, z]
Out[9]=	2 4 2 4 6 2 1 a 2 2 a 4 z 3 -6 - -- - 3 a - a - a + -- + ----- + -- - --- - --- + --- - 6 a z - 2 2 2 2 2 a z z a a z a z z 2 3 3 5 2 2 z 2 2 4 2 6 2 z z 3 > 2 a z + z - ---- + 6 a z + 6 a z + 3 a z + -- + -- + 13 a z + 2 3 a a a 4 5 3 3 5 3 4 5 z 2 4 4 4 6 4 z > 20 a z + 7 a z + 8 z + ---- + a z - 5 a z - 3 a z - -- - 2 a a 7 5 3 5 5 5 6 2 6 4 6 6 6 2 z > 16 a z - 24 a z - 9 a z - 6 z - 12 a z - 5 a z + a z + ---- + a 7 3 7 5 7 8 2 8 4 8 9 3 9 > 5 a z + 6 a z + 3 a z + 3 z + 6 a z + 3 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 1 2 1 4 3 5 3 5 - + 6 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 5 5 5 3 3 2 5 2 7 3 > ----- + ---- + --- + 2 q t + 4 q t + 3 q t + 4 q t + q t 3 2 3 q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n309

L11n308 L11n310