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L11n284 L11n286

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X5,12,6,13 X3849 X13,2,14,3 X14,7,15,8 X4,15,1,16 X21,18,22,19 X9,21,10,20 X19,5,20,10 X11,16,12,17 X17,22,18,11

Gauss Code:

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

Jones Polynomial:

- q-9 + 2q-8 - 3q-7 + 5q-6 - 4q-5 + 5q-4 - 3q-3 + 3q-2 - q-1 + 1

A2 (sl(3)) Invariant:

- q-32 + q-24 + 2q-20 + q-18 + 3q-16 + 4q-14 + 3q-12 + 5q-10 + 3q-8 + 3q-6 + q-4 + q-2 + 1

HOMFLY-PT Polynomial:

a2z-2 + 3a2 + 4a2z2 + a2z4 - 2a4z-2 - 4a4 - 4a4z2 - 4a4z4 - a4z6 + a6z-2 - 3a6z2 - 4a6z4 - a6z6 + 2a8 + 4a8z2 + a8z4 - a10

Kauffman Polynomial:

a2z-2 - 4a2 + 7a2z2 - 5a2z4 + a2z6 - 2a3z-1 + 4a3z - a3z3 - 3a3z5 + a3z7 + 2a4z-2 - 7a4 + 10a4z2 - 5a4z4 - 2a4z6 + a4z8 - 2a5z-1 + 6a5z - 8a5z3 + 6a5z5 - 4a5z7 + a5z9 + a6z-2 - 2a6 - 9a6z2 + 21a6z4 - 14a6z6 + 3a6z8 + 5a7z5 - 4a7z7 + a7z9 + 4a8 - 16a8z2 + 23a8z4 - 11a8z6 + 2a8z8 - 4a9z + 8a9z3 - 4a9z5 + a9z7 + 2a10 - 4a10z2 + 2a10z4 - 2a11z + a11z3

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 1                   1 j = -1                     j = -3               3 1   j = -5           1 2 1     j = -7           4 2       j = -9       1 3 3         j = -11       3 2           j = -13     1 3             j = -15   1 2               j = -17   1                 j = -19 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, 285]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 285]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 12, 6, 13], X[3, 8, 4, 9], X[13, 2, 14, 3], > X[14, 7, 15, 8], X[4, 15, 1, 16], X[21, 18, 22, 19], X[9, 21, 10, 20], > X[19, 5, 20, 10], X[11, 16, 12, 17], X[17, 22, 18, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 4, -3, -6}, {-2, -1, 5, 3, -8, 9}, > {-10, 2, -4, -5, 6, 10, -11, 7, -9, 8, -7, 11}]
In[6]:=	Jones[L][q]
Out[6]=	-9 2 3 5 4 5 3 3 1 1 - q + -- - -- + -- - -- + -- - -- + -- - - 8 7 6 5 4 3 2 q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 -24 2 -18 3 4 3 5 3 3 -4 -2 1 - q + q + --- + q + --- + --- + --- + --- + -- + -- + q + q 20 16 14 12 10 8 6 q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 285]][a, z]
Out[8]=	2 4 6 2 4 8 10 a 2 a a 2 2 4 2 6 2 3 a - 4 a + 2 a - a + -- - ---- + -- + 4 a z - 4 a z - 3 a z + 2 2 2 z z z 8 2 2 4 4 4 6 4 8 4 4 6 6 6 > 4 a z + a z - 4 a z - 4 a z + a z - a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 285]][a, z]
Out[9]=	2 4 6 3 5 2 4 6 8 10 a 2 a a 2 a 2 a 3 -4 a - 7 a - 2 a + 4 a + 2 a + -- + ---- + -- - ---- - ---- + 4 a z + 2 2 2 z z z z z 5 9 11 2 2 4 2 6 2 8 2 > 6 a z - 4 a z - 2 a z + 7 a z + 10 a z - 9 a z - 16 a z - 10 2 3 3 5 3 9 3 11 3 2 4 4 4 > 4 a z - a z - 8 a z + 8 a z + a z - 5 a z - 5 a z + 6 4 8 4 10 4 3 5 5 5 7 5 9 5 > 21 a z + 23 a z + 2 a z - 3 a z + 6 a z + 5 a z - 4 a z + 2 6 4 6 6 6 8 6 3 7 5 7 7 7 9 7 > a z - 2 a z - 14 a z - 11 a z + a z - 4 a z - 4 a z + a z + 4 8 6 8 8 8 5 9 7 9 > a z + 3 a z + 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-5 3 1 1 1 2 1 3 3 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 19 7 17 6 15 6 15 5 13 5 13 4 11 4 q q t q t q t q t q t q t q t 1 2 3 3 4 1 2 2 t 2 > ----- + ------ + ----- + ----- + ----- + ----- + ---- + ---- + -- + q t 9 4 11 3 9 3 9 2 7 2 5 2 7 5 3 q t q t q t q t q t q t q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n285

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