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L11n417 L11n419

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DrawMorseLink

PD Presentation:

X8192 X7,16,8,17 X3,10,4,11 X2,18,3,17 X18,9,19,10 X11,20,12,21 X5,14,6,15 X15,13,16,22 X13,6,14,1 X19,5,20,4 X21,12,22,7

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

q-28 + q-26 + q-24 + 3q-22 + q-20 + 3q-18 + 4q-16 + 3q-14 + 5q-12 + 2q-10 + 3q-8 + q-6 - q-4 + q-2 - 1

HOMFLY-PT Polynomial:

- a2 - 2a2z2 - a2z4 + a4z-2 + 7a4 + 10a4z2 + 5a4z4 + a4z6 - 2a6z-2 - 8a6 - 7a6z2 - 2a6z4 + a8z-2 + 2a8 + a8z2

Kauffman Polynomial:

- az + az3 + 2a2 - 4a2z2 + 3a2z4 - 3a3z + 6a3z3 - 2a3z5 + a3z7 - a4z-2 + 10a4 - 23a4z2 + 23a4z4 - 9a4z6 + 2a4z8 + 2a5z-1 - 8a5z + 6a5z3 + a5z5 - 2a5z7 + a5z9 - 2a6z-2 + 12a6 - 27a6z2 + 28a6z4 - 16a6z6 + 4a6z8 + 2a7z-1 - 6a7z + 6a7z3 - 4a7z5 - a7z7 + a7z9 - a8z-2 + 4a8 - 4a8z2 + 4a8z4 - 6a8z6 + 2a8z8 + 5a9z3 - 7a9z5 + 2a9z7 - a10 + 4a10z2 - 4a10z4 + a10z6

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 j = 1                   1 j = -1                 2   j = -3               3 2   j = -5             4 1     j = -7           3 4       j = -9         4 3         j = -11       2 4           j = -13     2 3             j = -15   1 3               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, 418]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 418]]
Out[4]=	PD[X[8, 1, 9, 2], X[7, 16, 8, 17], X[3, 10, 4, 11], X[2, 18, 3, 17], > X[18, 9, 19, 10], X[11, 20, 12, 21], X[5, 14, 6, 15], X[15, 13, 16, 22], > X[13, 6, 14, 1], X[19, 5, 20, 4], X[21, 12, 22, 7]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, -3, 10, -7, 9}, {-2, -1, 5, 3, -6, 11}, > {-9, 7, -8, 2, 4, -5, -10, 6, -11, 8}]
In[6]:=	Jones[L][q]
Out[6]=	-9 2 4 5 7 6 7 4 3 -1 + q - -- + -- - -- + -- - -- + -- - -- + - 8 7 6 5 4 3 2 q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 -26 -24 3 -20 3 4 3 5 2 3 -6 -1 + q + q + q + --- + q + --- + --- + --- + --- + --- + -- + q - 22 18 16 14 12 10 8 q q q q q q q -4 -2 > q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 418]][a, z]
Out[8]=	4 6 8 2 4 6 8 a 2 a a 2 2 4 2 6 2 -a + 7 a - 8 a + 2 a + -- - ---- + -- - 2 a z + 10 a z - 7 a z + 2 2 2 z z z 8 2 2 4 4 4 6 4 4 6 > a z - a z + 5 a z - 2 a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 418]][a, z]
Out[9]=	4 6 8 5 7 2 4 6 8 10 a 2 a a 2 a 2 a 2 a + 10 a + 12 a + 4 a - a - -- - ---- - -- + ---- + ---- - a z - 2 2 2 z z z z z 3 5 7 2 2 4 2 6 2 8 2 > 3 a z - 8 a z - 6 a z - 4 a z - 23 a z - 27 a z - 4 a z + 10 2 3 3 3 5 3 7 3 9 3 2 4 > 4 a z + a z + 6 a z + 6 a z + 6 a z + 5 a z + 3 a z + 4 4 6 4 8 4 10 4 3 5 5 5 7 5 > 23 a z + 28 a z + 4 a z - 4 a z - 2 a z + a z - 4 a z - 9 5 4 6 6 6 8 6 10 6 3 7 5 7 7 7 > 7 a z - 9 a z - 16 a z - 6 a z + a z + a z - 2 a z - a z + 9 7 4 8 6 8 8 8 5 9 7 9 > 2 a z + 2 a z + 4 a z + 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 1 1 3 2 3 2 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q q t q t q t q t q t q t q t 4 4 3 3 4 4 1 3 > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + q t 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q t q t q t q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n418

L11n417 L11n419