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L11n355 L11n357

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X11,18,12,19 X16,8,17,7 X8,16,9,15 X13,21,14,20 X19,22,20,15 X21,13,22,12 X17,14,18,5 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

- q-6 + 4q-5 - 6q-4 + 9q-3 - 9q-2 + 10q-1 - 7 + 6q - 3q2 + q3

A2 (sl(3)) Invariant:

- q-18 + 2q-16 + 2q-12 + 5q-10 + 2q-8 + 6q-6 + 2q-4 + 4q-2 + 3 + 2q4 - q6 + q10

HOMFLY-PT Polynomial:

a-2 + a-2z2 + z-2 - 4z2 - 2z4 - 2a2z-2 - 2a2 + 2a2z2 + 3a2z4 + a2z6 + a4z-2 + a4 - a4z2 - a4z4

Kauffman Polynomial:

- a-2 + 3a-2z2 - 3a-2z4 + a-2z6 + 6a-1z3 - 9a-1z5 + 3a-1z7 - z-2 + 2 + 3z2 - 2z4 - 6z6 + 3z8 + 2az-1 - 5az + 11az3 - 16az5 + 4az7 + az9 - 2a2z-2 + 6a2 - 6a2z2 + 5a2z4 - 10a2z6 + 5a2z8 + 2a3z-1 - 5a3z + 4a3z3 - 5a3z5 + 2a3z7 + a3z9 - a4z-2 + 4a4 - 8a4z2 + 8a4z4 - 3a4z6 + 2a4z8 + 2a5z5 + a5z7 - 2a6z2 + 4a6z4 + a7z3

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                   1 j = 5                 2   j = 3               4 1   j = 1             4 3     j = -1           6 3       j = -3         4 5         j = -5       5 5           j = -7     2 5             j = -9   2 4               j = -11   3                 j = -13 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, 356]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 356]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[11, 18, 12, 19], X[16, 8, 17, 7], > X[8, 16, 9, 15], X[13, 21, 14, 20], X[19, 22, 20, 15], X[21, 13, 22, 12], > X[17, 14, 18, 5], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {5, -4, -9, 3, -7, 6, -8, 7}, > {10, -1, 4, -5, 11, -2, -3, 8, -6, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-6 4 6 9 9 10 2 3 -7 - q + -- - -- + -- - -- + -- + 6 q - 3 q + q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 2 2 5 2 6 2 4 4 6 10 3 - q + --- + --- + --- + -- + -- + -- + -- + 2 q - q + q 16 12 10 8 6 4 2 q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 356]][a, z]
Out[8]=	2 4 2 -2 2 4 -2 2 a a 2 z 2 2 4 2 4 a - 2 a + a + z - ---- + -- - 4 z + -- + 2 a z - a z - 2 z + 2 2 2 z z a 2 4 4 4 2 6 > 3 a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 356]][a, z]
Out[9]=	2 4 3 -2 2 4 -2 2 a a 2 a 2 a 3 2 2 - a + 6 a + 4 a - z - ---- - -- + --- + ---- - 5 a z - 5 a z + 3 z + 2 2 z z z z 2 3 3 z 2 2 4 2 6 2 6 z 3 3 3 7 3 > ---- - 6 a z - 8 a z - 2 a z + ---- + 11 a z + 4 a z + a z - 2 a a 4 5 4 3 z 2 4 4 4 6 4 9 z 5 3 5 > 2 z - ---- + 5 a z + 8 a z + 4 a z - ---- - 16 a z - 5 a z + 2 a a 6 7 5 5 6 z 2 6 4 6 3 z 7 3 7 > 2 a z - 6 z + -- - 10 a z - 3 a z + ---- + 4 a z + 2 a z + 2 a a 5 7 8 2 8 4 8 9 3 9 > a z + 3 z + 5 a z + 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 6 1 3 2 4 2 5 5 5 -- + - + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 3 q 13 5 11 4 9 4 9 3 7 3 7 2 5 2 5 q q t q t q t q t q t q t q t q t 4 3 t 2 3 2 3 3 5 3 7 4 > ---- + --- + 4 q t + 3 q t + 4 q t + q t + 2 q t + q t 3 q q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n356

L11n355 L11n357