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L11n229 L11n231

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DrawMorseLink

PD Presentation:

X10,1,11,2 X11,17,12,16 X5,21,6,20 X3,12,4,13 X7,14,8,15 X13,6,14,7 X17,9,18,22 X21,19,22,18 X8,9,1,10 X19,5,20,4 X15,2,16,3

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 2q-11/2 - 5q-9/2 + 8q-7/2 - 10q-5/2 + 10q-3/2 - 10q-1/2 + 7q1/2 - 5q3/2 + 2q5/2

A2 (sl(3)) Invariant:

q-20 + q-16 + 3q-14 - q-12 + 2q-10 - q-6 + 2q-4 - q-2 + 4 + q6 - 2q8

HOMFLY-PT Polynomial:

3a-1z + 2a-1z3 - 5az - 6az3 - 2az5 - a3z-1 - 2a3z - 2a3z3 - a3z5 + a5z-1 + 2a5z + a5z3

Kauffman Polynomial:

4a-2z2 - 3a-2z4 - 3a-1z + 8a-1z3 - 3a-1z5 - a-1z7 + 4z2 - 3z4 + 2z6 - 2z8 - 2az + 3az5 - 2az7 - az9 + 4a2z2 - 11a2z4 + 10a2z6 - 5a2z8 - a3z-1 + 7a3z - 16a3z3 + 13a3z5 - 4a3z7 - a3z9 + a4 + 3a4z2 - 7a4z4 + 6a4z6 - 3a4z8 - a5z-1 + 4a5z - 5a5z3 + 6a5z5 - 3a5z7 - a6z2 + 4a6z4 - 2a6z6 - 2a7z + 3a7z3 - a7z5

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 6                   2 j = 4                 3   j = 2               4 2   j = 0             6 3     j = -2           5 5       j = -4         5 5         j = -6       3 5           j = -8     2 5             j = -10   1 4               j = -12   1                 j = -14 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, 230]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 230]]
Out[4]=	PD[X[10, 1, 11, 2], X[11, 17, 12, 16], X[5, 21, 6, 20], X[3, 12, 4, 13], > X[7, 14, 8, 15], X[13, 6, 14, 7], X[17, 9, 18, 22], X[21, 19, 22, 18], > X[8, 9, 1, 10], X[19, 5, 20, 4], X[15, 2, 16, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 11, -4, 10, -3, 6, -5, -9}, > {9, -1, -2, 4, -6, 5, -11, 2, -7, 8, -10, 3, -8, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 2 5 8 10 10 10 3/2 -q + ----- - ---- + ---- - ---- + ---- - ------- + 7 Sqrt[q] - 5 q + 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 5/2 > 2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -16 3 -12 2 -6 2 -2 6 8 4 + q + q + --- - q + --- - q + -- - q + q - 2 q 14 10 4 q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 230]][a, z]
Out[8]=	3 5 3 a a 3 z 3 5 2 z 3 3 3 5 3 -(--) + -- + --- - 5 a z - 2 a z + 2 a z + ---- - 6 a z - 2 a z + a z - z z a a 5 3 5 > 2 a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 230]][a, z]
Out[9]=	3 5 2 4 a a 3 z 3 5 7 2 4 z 2 2 a - -- - -- - --- - 2 a z + 7 a z + 4 a z - 2 a z + 4 z + ---- + 4 a z + z z a 2 a 3 4 4 2 6 2 8 z 3 3 5 3 7 3 4 3 z > 3 a z - a z + ---- - 16 a z - 5 a z + 3 a z - 3 z - ---- - a 2 a 5 2 4 4 4 6 4 3 z 5 3 5 5 5 7 5 > 11 a z - 7 a z + 4 a z - ---- + 3 a z + 13 a z + 6 a z - a z + a 7 6 2 6 4 6 6 6 z 7 3 7 5 7 > 2 z + 10 a z + 6 a z - 2 a z - -- - 2 a z - 4 a z - 3 a z - a 8 2 8 4 8 9 3 9 > 2 z - 5 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 1 1 1 4 2 5 3 5 6 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 5 5 5 2 2 2 4 2 6 3 > ----- + ---- + ---- + 3 t + 4 q t + 2 q t + 3 q t + 2 q t 4 2 4 2 q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n230

L11n229 L11n231