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L11n382 L11n384

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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X4,15,1,16 X10,6,11,5 X8493 X17,22,18,19 X11,20,12,21 X19,12,20,13 X21,18,22,5 X16,10,17,9 X2,14,3,13

Gauss Code:

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

Jones Polynomial:

- q-7 + 3q-6 - 3q-5 + 5q-4 - 4q-3 + 6q-2 - 4q-1 + 3 - 2q + q2

A2 (sl(3)) Invariant:

- q-22 + 2q-18 + 3q-16 + 6q-14 + 5q-12 + 5q-10 + 4q-8 + q-6 + q-4 - q-2 + q2 + q6

HOMFLY-PT Polynomial:

2 + 3z2 + z4 + a2z-2 - 3a2 - 8a2z2 - 5a2z4 - a2z6 - 2a4z-2 + 2a4 + 6a4z2 + 2a4z4 + a6z-2 - a6 - a6z2

Kauffman Polynomial:

a-2z2 - 2a-1z + 2a-1z3 + 3 - 6z2 + 3z4 - 7az + 11az3 - 8az5 + 2az7 + a2z-2 + 5a2 - 26a2z2 + 36a2z4 - 21a2z6 + 4a2z8 - 2a3z-1 - 5a3z + 17a3z3 - 5a3z5 - 6a3z7 + 2a3z9 + 2a4z-2 + 2a4 - 27a4z2 + 53a4z4 - 36a4z6 + 7a4z8 - 2a5z-1 - a5z + 11a5z3 - a5z5 - 7a5z7 + 2a5z9 + a6z-2 - a6 - 8a6z2 + 20a6z4 - 15a6z6 + 3a6z8 - a7z + 3a7z3 - 4a7z5 + a7z7

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 5                   1 j = 3                 2 1 j = 1               2 1   j = -1             4 4 1   j = -3           3 3 2     j = -5         3 5         j = -7       3 3 1         j = -9     2 5 1           j = -11   1 1               j = -13   2                 j = -15 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, 383]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 383]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[10, 6, 11, 5], > X[8, 4, 9, 3], X[17, 22, 18, 19], X[11, 20, 12, 21], X[19, 12, 20, 13], > X[21, 18, 22, 5], X[16, 10, 17, 9], X[2, 14, 3, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {-8, 7, -9, 6}, > {4, -1, 2, -5, 10, -4, -7, 8, 11, -2, 3, -10, -6, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-7 3 3 5 4 6 4 2 3 - q + -- - -- + -- - -- + -- - - - 2 q + q 6 5 4 3 2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 2 3 6 5 5 4 -6 -4 -2 2 6 -q + --- + --- + --- + --- + --- + -- + q + q - q + q + q 18 16 14 12 10 8 q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 383]][a, z]
Out[8]=	2 4 6 2 4 6 a 2 a a 2 2 2 4 2 6 2 4 2 - 3 a + 2 a - a + -- - ---- + -- + 3 z - 8 a z + 6 a z - a z + z - 2 2 2 z z z 2 4 4 4 2 6 > 5 a z + 2 a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 383]][a, z]
Out[9]=	2 4 6 3 5 2 4 6 a 2 a a 2 a 2 a 2 z 3 3 + 5 a + 2 a - a + -- + ---- + -- - ---- - ---- - --- - 7 a z - 5 a z - 2 2 2 z z a z z z 2 3 5 7 2 z 2 2 4 2 6 2 2 z 3 > a z - a z - 6 z + -- - 26 a z - 27 a z - 8 a z + ---- + 11 a z + 2 a a 3 3 5 3 7 3 4 2 4 4 4 6 4 > 17 a z + 11 a z + 3 a z + 3 z + 36 a z + 53 a z + 20 a z - 5 3 5 5 5 7 5 2 6 4 6 6 6 > 8 a z - 5 a z - a z - 4 a z - 21 a z - 36 a z - 15 a z + 7 3 7 5 7 7 7 2 8 4 8 6 8 > 2 a z - 6 a z - 7 a z + a z + 4 a z + 7 a z + 3 a z + 3 9 5 9 > 2 a z + 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 4 1 2 1 1 2 5 3 -- + - + 2 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 3 q 15 7 13 6 11 6 11 5 9 5 9 4 7 4 q q t q t q t q t q t q t q t 1 3 3 1 5 3 3 4 t > ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + - + q t + 9 3 7 3 5 3 7 2 5 2 3 2 3 q t q q t q t q t q t q t q t q t 3 3 2 5 2 > 2 q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n383

L11n382 L11n384