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L11n426 L11n428

Knotscape

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DrawMorseLink

PD Presentation:

X8192 X11,18,12,19 X3,10,4,11 X19,2,20,3 X7,16,8,17 X20,9,21,10 X17,12,18,7 X22,16,13,15 X14,6,15,5 X4,14,5,13 X6,21,1,22

Gauss Code:

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

Jones Polynomial:

2q-7 - 3q-6 + 7q-5 - 8q-4 + 10q-3 - 9q-2 + 8q-1 - 5 + 3q - q2

A2 (sl(3)) Invariant:

q-26 + q-24 + 3q-22 + 3q-20 + 3q-18 + 6q-16 + 2q-14 + 4q-12 + 2q-10 + 2q-6 - 2q-4 + 2q-2 + q4 - q6

HOMFLY-PT Polynomial:

- 2z2 - z4 + 2a2z2 + 3a2z4 + a2z6 + a4z-2 + 4a4 + 6a4z2 + 4a4z4 + a4z6 - 2a6z-2 - 5a6 - 4a6z2 - a6z4 + a8z-2 + a8

Kauffman Polynomial:

- 2a-1z3 + a-1z5 + 3z2 - 7z4 + 3z6 - az + 5az3 - 9az5 + 4az7 + a2 - 2a2z2 + 3a2z4 - 5a2z6 + 3a2z8 - 3a3z + 11a3z3 - 10a3z5 + 3a3z7 + a3z9 - a4z-2 + 8a4 - 22a4z2 + 27a4z4 - 15a4z6 + 5a4z8 + 2a5z-1 - 8a5z + 7a5z3 - a5z5 + a5z9 - 2a6z-2 + 12a6 - 25a6z2 + 20a6z4 - 7a6z6 + 2a6z8 + 2a7z-1 - 6a7z + 3a7z3 - a7z5 + a7z7 - a8z-2 + 6a8 - 8a8z2 + 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                 2   j = 1               3 1   j = -1             5 2     j = -3           5 4       j = -5         5 4         j = -7       4 6           j = -9     3 4             j = -11   1 5               j = -13 1 2                 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, 427]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 427]]
Out[4]=	PD[X[8, 1, 9, 2], X[11, 18, 12, 19], X[3, 10, 4, 11], X[19, 2, 20, 3], > X[7, 16, 8, 17], X[20, 9, 21, 10], X[17, 12, 18, 7], X[22, 16, 13, 15], > X[14, 6, 15, 5], X[4, 14, 5, 13], X[6, 21, 1, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 4, -3, -10, 9, -11}, {-5, -1, 6, 3, -2, 7}, > {10, -9, 8, 5, -7, 2, -4, -6, 11, -8}]
In[6]:=	Jones[L][q]
Out[6]=	2 3 7 8 10 9 8 2 -5 + -- - -- + -- - -- + -- - -- + - + 3 q - q 7 6 5 4 3 2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 -24 3 3 3 6 2 4 2 2 2 2 4 6 q + q + --- + --- + --- + --- + --- + --- + --- + -- - -- + -- + q - q 22 20 18 16 14 12 10 6 4 2 q q q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 427]][a, z]
Out[8]=	4 6 8 4 6 8 a 2 a a 2 2 2 4 2 6 2 4 4 a - 5 a + a + -- - ---- + -- - 2 z + 2 a z + 6 a z - 4 a z - z + 2 2 2 z z z 2 4 4 4 6 4 2 6 4 6 > 3 a z + 4 a z - a z + a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 427]][a, z]
Out[9]=	4 6 8 5 7 2 4 6 8 a 2 a a 2 a 2 a 3 a + 8 a + 12 a + 6 a - -- - ---- - -- + ---- + ---- - a z - 3 a z - 2 2 2 z z z z z 3 5 7 2 2 2 4 2 6 2 8 2 2 z > 8 a z - 6 a z + 3 z - 2 a z - 22 a z - 25 a z - 8 a z - ---- + a 3 3 3 5 3 7 3 4 2 4 4 4 > 5 a z + 11 a z + 7 a z + 3 a z - 7 z + 3 a z + 27 a z + 5 6 4 8 4 z 5 3 5 5 5 7 5 6 > 20 a z + 3 a z + -- - 9 a z - 10 a z - a z - a z + 3 z - a 2 6 4 6 6 6 7 3 7 7 7 2 8 > 5 a z - 15 a z - 7 a z + 4 a z + 3 a z + a z + 3 a z + 4 8 6 8 3 9 5 9 > 5 a z + 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 5 2 1 2 1 5 3 4 4 -- + - + ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 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 6 5 4 5 2 t 2 3 2 5 3 > ----- + ----- + ---- + ---- + --- + 3 q t + q t + 2 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 L11n427

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