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L11n391 L11n393

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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X5,12,6,13 X8493 X22,14,19,13 X20,10,21,9 X10,20,11,19 X14,22,15,21 X11,18,12,5 X2,16,3,15

Gauss Code:

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

Jones Polynomial:

- q-5 + 4q-4 - 9q-3 + 13q-2 - 16q-1 + 17 - 14q + 13q2 - 6q3 + 3q4

A2 (sl(3)) Invariant:

- q-16 + q-14 + 2q-12 - 4q-10 + q-8 - 3q-6 - 4q-4 + 2q-2 - 2 + 7q2 + 3q4 + 7q6 + 9q8 + 2q10 + 5q12 + 2q14

HOMFLY-PT Polynomial:

2a-4z-2 + 2a-4 - 5a-2z-2 - 8a-2 - 3a-2z2 + a-2z4 + 4z-2 + 8 + 3z2 - z4 - z6 - a2z-2 - 2a2 + a2z2 + 2a2z4 - a4z2

Kauffman Polynomial:

- 2a-4z-2 + 8a-4 - 11a-4z2 + 6a-4z4 + 5a-3z-1 - 13a-3z + 10a-3z3 - 3a-3z5 + 3a-3z7 - 5a-2z-2 + 16a-2 - 24a-2z2 + 17a-2z4 - 8a-2z6 + 5a-2z8 + 9a-1z-1 - 24a-1z + 29a-1z3 - 26a-1z5 + 10a-1z7 + 2a-1z9 - 4z-2 + 11 - 15z2 + 12z4 - 18z6 + 12z8 + 5az-1 - 14az + 29az3 - 38az5 + 15az7 + 2az9 - a2z-2 + 2a2 - 4a2z4 - 6a2z6 + 7a2z8 + a3z-1 - 3a3z + 9a3z3 - 14a3z5 + 8a3z7 + 2a4z2 - 5a4z4 + 4a4z6 - a5z3 + a5z5

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 9                   3 j = 7                 5 2 j = 5               8 1   j = 3             6 5     j = 1           11 8       j = -1         7 8         j = -3       6 9           j = -5     3 7             j = -7   1 6               j = -9   3                 j = -11 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, 392]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 392]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[5, 12, 6, 13], > X[8, 4, 9, 3], X[22, 14, 19, 13], X[20, 10, 21, 9], X[10, 20, 11, 19], > X[14, 22, 15, 21], X[11, 18, 12, 5], X[2, 16, 3, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {8, -7, 9, -6}, > {-4, -1, 2, -5, 7, -8, -10, 4, 6, -9, 11, -2, 3, 10}]
In[6]:=	Jones[L][q]
Out[6]=	-5 4 9 13 16 2 3 4 17 - q + -- - -- + -- - -- - 14 q + 13 q - 6 q + 3 q 4 3 2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -14 2 4 -8 3 4 2 2 4 6 8 -2 - q + q + --- - --- + q - -- - -- + -- + 7 q + 3 q + 7 q + 9 q + 12 10 6 4 2 q q q q q 10 12 14 > 2 q + 5 q + 2 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 392]][a, z]
Out[8]=	2 2 2 8 2 4 2 5 a 2 3 z 2 2 4 2 8 + -- - -- - 2 a + -- + ----- - ----- - -- + 3 z - ---- + a z - a z - 4 2 2 4 2 2 2 2 2 a a z a z a z z a 4 4 z 2 4 6 > z + -- + 2 a z - z 2 a
In[9]:=	Kauffman[Link[11, NonAlternating, 392]][a, z]
Out[9]=	2 3 8 16 2 4 2 5 a 5 9 5 a a 13 z 11 + -- + -- + 2 a - -- - ----- - ----- - -- + ---- + --- + --- + -- - ---- - 4 2 2 4 2 2 2 2 3 a z z z 3 a a z a z a z z a z a 2 2 3 3 24 z 3 2 11 z 24 z 4 2 10 z 29 z > ---- - 14 a z - 3 a z - 15 z - ----- - ----- + 2 a z + ----- + ----- + a 4 2 3 a a a a 4 4 3 3 3 5 3 4 6 z 17 z 2 4 4 4 > 29 a z + 9 a z - a z + 12 z + ---- + ----- - 4 a z - 5 a z - 4 2 a a 5 5 6 3 z 26 z 5 3 5 5 5 6 8 z 2 6 > ---- - ----- - 38 a z - 14 a z + a z - 18 z - ---- - 6 a z + 3 a 2 a a 7 7 8 4 6 3 z 10 z 7 3 7 8 5 z 2 8 > 4 a z + ---- + ----- + 15 a z + 8 a z + 12 z + ---- + 7 a z + 3 a 2 a a 9 2 z 9 > ---- + 2 a z a
In[10]:=	Kh[L][q, t]
Out[10]=	8 1 3 1 6 3 7 6 9 - + 11 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t 7 3 3 2 5 2 5 3 7 3 7 4 > --- + 8 q t + 6 q t + 5 q t + 8 q t + q t + 5 q t + 2 q t + q t 9 4 > 3 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n392

L11n391 L11n393