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L11n316 L11n318

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X16,7,17,8 X9,20,10,21 X11,18,12,19 X19,22,20,11 X8,15,9,16 X21,10,22,5 X14,18,15,17 X2536 X4,14,1,13

Gauss Code:

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

Jones Polynomial:

2q-7 - 5q-6 + 9q-5 - 11q-4 + 14q-3 - 12q-2 + 11q-1 - 7 + 4q - q2

A2 (sl(3)) Invariant:

2q-22 + 5q-16 + q-14 + 5q-12 + 4q-10 + 2q-8 + 5q-6 - q-4 + 4q-2 - q2 + 2q4 - q6

HOMFLY-PT Polynomial:

- z2 - z4 + a2z-2 + 4a2 + 5a2z2 + 3a2z4 + a2z6 - 2a4z-2 - 6a4 - 7a4z2 - 3a4z4 + a6z-2 + 2a6 + 2a6z2

Kauffman Polynomial:

- a-1z3 + a-1z5 + 3z2 - 7z4 + 4z6 + 3az3 - 10az5 + 6az7 + a2z-2 - 5a2 + 11a2z2 - 14a2z4 + a2z6 + 4a2z8 - 2a3z-1 + 6a3z + 4a3z3 - 19a3z5 + 10a3z7 + a3z9 + 2a4z-2 - 8a4 + 16a4z2 - 13a4z4 - 2a4z6 + 6a4z8 - 2a5z-1 + 6a5z - 4a5z3 - 5a5z5 + 5a5z7 + a5z9 + a6z-2 - 3a6 + 4a6z2 - 3a6z4 + a6z6 + 2a6z8 - 4a7z3 + 3a7z5 + a7z7 + a8 - 4a8z2 + 3a8z4

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 5                   1 j = 3                 3   j = 1               4 1   j = -1             7 3     j = -3           7 6       j = -5         7 5         j = -7       5 8           j = -9     4 6             j = -11   1 5               j = -13 1 4                 j = -15 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, 317]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 317]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[16, 7, 17, 8], X[9, 20, 10, 21], > X[11, 18, 12, 19], X[19, 22, 20, 11], X[8, 15, 9, 16], X[21, 10, 22, 5], > X[14, 18, 15, 17], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -7, -4, 8}, > {-5, -2, 11, -9, 7, -3, 9, 5, -6, 4, -8, 6}]
In[6]:=	Jones[L][q]
Out[6]=	2 5 9 11 14 12 11 2 -7 + -- - -- + -- - -- + -- - -- + -- + 4 q - q 7 6 5 4 3 2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 5 -14 5 4 2 5 -4 4 2 4 6 --- + --- + q + --- + --- + -- + -- - q + -- - q + 2 q - q 22 16 12 10 8 6 2 q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 317]][a, z]
Out[8]=	2 4 6 2 4 6 a 2 a a 2 2 2 4 2 6 2 4 4 a - 6 a + 2 a + -- - ---- + -- - z + 5 a z - 7 a z + 2 a z - z + 2 2 2 z z z 2 4 4 4 2 6 > 3 a z - 3 a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 317]][a, z]
Out[9]=	2 4 6 3 5 2 4 6 8 a 2 a a 2 a 2 a 3 5 -5 a - 8 a - 3 a + a + -- + ---- + -- - ---- - ---- + 6 a z + 6 a z + 2 2 2 z z z z z 3 2 2 2 4 2 6 2 8 2 z 3 3 3 > 3 z + 11 a z + 16 a z + 4 a z - 4 a z - -- + 3 a z + 4 a z - a 5 5 3 7 3 4 2 4 4 4 6 4 8 4 z > 4 a z - 4 a z - 7 z - 14 a z - 13 a z - 3 a z + 3 a z + -- - a 5 3 5 5 5 7 5 6 2 6 4 6 6 6 > 10 a z - 19 a z - 5 a z + 3 a z + 4 z + a z - 2 a z + a z + 7 3 7 5 7 7 7 2 8 4 8 6 8 3 9 > 6 a z + 10 a z + 5 a z + a z + 4 a z + 6 a z + 2 a z + a z + 5 9 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	6 7 2 1 4 1 5 4 6 5 -- + - + ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 3 q 15 6 13 6 13 5 11 5 11 4 9 4 9 3 7 3 q q t q t q t q t q t q t q t q t 8 7 5 7 3 t 2 3 2 5 3 > ----- + ----- + ---- + ---- + --- + 4 q t + q t + 3 q t + q t 7 2 5 2 5 3 q q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n317

L11n316 L11n318