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L11n197 L11n199

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DrawMorseLink

PD Presentation:

X10,1,11,2 X12,3,13,4 X5,14,6,15 X7,17,8,16 X15,21,16,20 X18,14,19,13 X21,6,22,7 X22,18,9,17 X19,5,20,4 X2,9,3,10 X8,11,1,12

Gauss Code:

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

Jones Polynomial:

- q-9/2 + q-7/2 - 3q-5/2 + 4q-3/2 - 5q-1/2 + 4q1/2 - 4q3/2 + 3q5/2 - 2q7/2 + q9/2

A2 (sl(3)) Invariant:

q-14 + q-12 + 2q-10 + 3q-8 + q-6 + 2q-4 - q4 + q6 - q14

HOMFLY-PT Polynomial:

2a-3z + a-3z3 + a-1z-1 - a-1z - 3a-1z3 - a-1z5 - 3az-1 - 6az - 4az3 - az5 + 2a3z-1 + 3a3z + a3z3

Kauffman Polynomial:

- 3a-4z2 + 4a-4z4 - a-4z6 + 2a-3z - 7a-3z3 + 8a-3z5 - 2a-3z7 - a-2 + 4a-2z2 - 8a-2z4 + 8a-2z6 - 2a-2z8 + a-1z-1 - a-1z - a-1z5 + 3a-1z7 - a-1z9 - 3 + 14z2 - 22z4 + 14z6 - 3z8 + 3az-1 - 11az + 13az3 - 11az5 + 5az7 - az9 - 3a2 + 8a2z2 - 11a2z4 + 5a2z6 - a2z8 + 2a3z-1 - 6a3z + 5a3z3 - 2a3z5 + a4z2 - a4z4 + 2a5z - a5z3

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 10                   1 j = 8                 1   j = 6               2 1   j = 4             2 1     j = 2           2 2       j = 0         3 2         j = -2       2 3           j = -4     1 2             j = -6     2               j = -8 1 1                 j = -10 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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 198]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 198]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 3, 13, 4], X[5, 14, 6, 15], X[7, 17, 8, 16], > X[15, 21, 16, 20], X[18, 14, 19, 13], X[21, 6, 22, 7], X[22, 18, 9, 17], > X[19, 5, 20, 4], X[2, 9, 3, 10], X[8, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, 9, -3, 7, -4, -11}, > {10, -1, 11, -2, 6, 3, -5, 4, 8, -6, -9, 5, -7, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) -(7/2) 3 4 5 3/2 5/2 -q + q - ---- + ---- - ------- + 4 Sqrt[q] - 4 q + 3 q - 5/2 3/2 Sqrt[q] q q 7/2 9/2 > 2 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -12 2 3 -6 2 4 6 14 q + q + --- + -- + q + -- - q + q - q 10 8 4 q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 198]][a, z]
Out[8]=	3 3 3 5 1 3 a 2 a 2 z z 3 z 3 z 3 3 3 z --- - --- + ---- + --- - - - 6 a z + 3 a z + -- - ---- - 4 a z + a z - -- - a z z z 3 a 3 a a a a 5 > a z
In[9]:=	Kauffman[Link[11, NonAlternating, 198]][a, z]
Out[9]=	3 -2 2 1 3 a 2 a 2 z z 3 5 -3 - a - 3 a + --- + --- + ---- + --- - - - 11 a z - 6 a z + 2 a z + a z z z 3 a a 2 2 3 2 3 z 4 z 2 2 4 2 7 z 3 3 3 5 3 > 14 z - ---- + ---- + 8 a z + a z - ---- + 13 a z + 5 a z - a z - 4 2 3 a a a 4 4 5 5 4 4 z 8 z 2 4 4 4 8 z z 5 3 5 > 22 z + ---- - ---- - 11 a z - a z + ---- - -- - 11 a z - 2 a z + 4 2 3 a a a a 6 6 7 7 8 6 z 8 z 2 6 2 z 3 z 7 8 2 z 2 8 > 14 z - -- + ---- + 5 a z - ---- + ---- + 5 a z - 3 z - ---- - a z - 4 2 3 a 2 a a a a 9 z 9 > -- - a z a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 1 2 1 2 2 2 3 + -- + ------ + ----- + ----- + ----- + ----- + ---- + ---- + 2 t + 2 q t + 2 10 4 8 4 8 3 6 2 4 2 4 2 q q t q t q t q t q t q t q t 2 2 4 2 4 3 6 3 6 4 8 4 10 5 > 2 q t + 2 q t + q t + 2 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n198

L11n197 L11n199