© | Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table:

L11n388 L11n390

Knotscape

This page is passe. Go here instead! The 3-Component Link L11n389 Visit L11n389's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X5,12,6,13 X3849 X13,22,14,19 X9,20,10,21 X19,10,20,11 X21,14,22,15 X11,18,12,5 X15,2,16,3

Gauss Code:

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

Jones Polynomial:

- q-12 + 3q-11 - 6q-10 + 8q-9 - 7q-8 + 9q-7 - 6q-6 + 5q-5 - 2q-4 + q-3

A2 (sl(3)) Invariant:

- q-42 - 2q-40 - q-38 - 2q-36 + q-34 + q-32 + 3q-30 + 6q-28 + 4q-26 + 8q-24 + 3q-22 + 3q-20 + 2q-18 + 2q-14 - q-12 + q-10

HOMFLY-PT Polynomial:

a6 + 4a6z2 + 4a6z4 + a6z6 + 2a8z-2 + 8a8 + 12a8z2 + 9a8z4 + 2a8z6 - 5a10z-2 - 15a10 - 13a10z2 - 3a10z4 + 4a12z-2 + 6a12 + a12z2 - a14z-2

Kauffman Polynomial:

- a6 + 4a6z2 - 4a6z4 + a6z6 + 3a7z3 - 6a7z5 + 2a7z7 - 2a8z-2 + 9a8 - 16a8z2 + 15a8z4 - 11a8z6 + 3a8z8 + 5a9z-1 - 20a9z + 30a9z3 - 21a9z5 + 3a9z7 + a9z9 - 5a10z-2 + 21a10 - 42a10z2 + 42a10z4 - 24a10z6 + 6a10z8 + 9a11z-1 - 33a11z + 44a11z3 - 22a11z5 + 3a11z7 + a11z9 - 4a12z-2 + 14a12 - 25a12z2 + 26a12z4 - 12a12z6 + 3a12z8 + 5a13z-1 - 15a13z + 18a13z3 - 7a13z5 + 2a13z7 - a14z-2 + 2a14 - 3a14z2 + 3a14z4 + a15z-1 - 2a15z + a15z3

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -5                   1 j = -7                 2 1 j = -9               3     j = -11             3 2     j = -13           6 3       j = -15         3 5         j = -17       5 4           j = -19     1 3             j = -21   2 5               j = -23   2                 j = -25 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, 389]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 389]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[5, 12, 6, 13], > X[3, 8, 4, 9], X[13, 22, 14, 19], X[9, 20, 10, 21], X[19, 10, 20, 11], > X[21, 14, 22, 15], X[11, 18, 12, 5], X[15, 2, 16, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 11, -5, -3}, {-8, 7, -9, 6}, > {-4, -1, 2, 5, -7, 8, -10, 4, -6, 9, -11, -2, 3, 10}]
In[6]:=	Jones[L][q]
Out[6]=	-12 3 6 8 7 9 6 5 2 -3 -q + --- - --- + -- - -- + -- - -- + -- - -- + q 11 10 9 8 7 6 5 4 q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 2 -38 2 -34 -32 3 6 4 8 3 3 -q - --- - q - --- + q + q + --- + --- + --- + --- + --- + --- + 40 36 30 28 26 24 22 20 q q q q q q q q 2 2 -12 -10 > --- + --- - q + q 18 14 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 389]][a, z]
Out[8]=	8 10 12 14 6 8 10 12 2 a 5 a 4 a a 6 2 8 2 a + 8 a - 15 a + 6 a + ---- - ----- + ----- - --- + 4 a z + 12 a z - 2 2 2 2 z z z z 10 2 12 2 6 4 8 4 10 4 6 6 8 6 > 13 a z + a z + 4 a z + 9 a z - 3 a z + a z + 2 a z
In[9]:=	Kauffman[Link[11, NonAlternating, 389]][a, z]
Out[9]=	8 10 12 14 9 6 8 10 12 14 2 a 5 a 4 a a 5 a -a + 9 a + 21 a + 14 a + 2 a - ---- - ----- - ----- - --- + ---- + 2 2 2 2 z z z z z 11 13 15 9 a 5 a a 9 11 13 15 6 2 > ----- + ----- + --- - 20 a z - 33 a z - 15 a z - 2 a z + 4 a z - z z z 8 2 10 2 12 2 14 2 7 3 9 3 > 16 a z - 42 a z - 25 a z - 3 a z + 3 a z + 30 a z + 11 3 13 3 15 3 6 4 8 4 10 4 > 44 a z + 18 a z + a z - 4 a z + 15 a z + 42 a z + 12 4 14 4 7 5 9 5 11 5 13 5 6 6 > 26 a z + 3 a z - 6 a z - 21 a z - 22 a z - 7 a z + a z - 8 6 10 6 12 6 7 7 9 7 11 7 > 11 a z - 24 a z - 12 a z + 2 a z + 3 a z + 3 a z + 13 7 8 8 10 8 12 8 9 9 11 9 > 2 a z + 3 a z + 6 a z + 3 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-7 -5 1 2 2 5 1 3 5 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 25 9 23 8 21 8 21 7 19 7 19 6 17 6 q t q t q t q t q t q t q t 4 3 5 6 3 3 2 3 2 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ---- 17 5 15 5 15 4 13 4 13 3 11 3 11 2 9 2 7 q t q t q t q t q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n389

L11n388 L11n390