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

L11n436 L11n438

Knotscape

This page is passe. Go here instead! The 3-Component Link L11n437 Visit L11n437's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X8192 X12,3,13,4 X13,20,14,21 X9,18,10,19 X21,10,22,11 X19,14,20,7 X5,17,6,16 X17,22,18,15 X2738 X4,11,5,12 X15,1,16,6

Gauss Code:

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

Jones Polynomial:

q-9 - q-8 + q-7 + q-5 + q-4 + q-2 - q-1 + 1

A2 (sl(3)) Invariant:

q-28 + 2q-24 + 3q-22 + 3q-20 + 4q-18 + 3q-16 + 4q-14 + 2q-12 + 2q-10 + q-8 + q-2 + 1

HOMFLY-PT Polynomial:

2a2 + 4a2z2 + a2z4 + a4z-2 - 5a4z2 - 5a4z4 - a4z6 - 2a6z-2 - 3a6 + a8z-2 + a8 + a8z2

Kauffman Polynomial:

- 2a2 + 6a2z2 - 5a2z4 + a2z6 + 5a3z3 - 5a3z5 + a3z7 - a4z-2 + 3a4 + 3a4z2 - 5a4z4 + a4z6 + 2a5z-1 - 9a5z + 13a5z3 - 7a5z5 + a5z7 - 2a6z-2 + 11a6 - 25a6z2 + 21a6z4 - 8a6z6 + a6z8 + 2a7z-1 - 9a7z + 13a7z3 - 7a7z5 + a7z7 - a8z-2 + 7a8 - 17a8z2 + 16a8z4 - 7a8z6 + a8z8 + 5a9z3 - 5a9z5 + a9z7 + 5a10z2 - 5a10z4 + a10z6

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 1                     1 j = -1                       j = -3               1 2 1   j = -5             1 1 1     j = -7           2 4 1       j = -9         1 1 2         j = -11       1 3 1           j = -13     1 1 1             j = -15     1 1               j = -17 1 1                   j = -19 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, 437]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 437]]
Out[4]=	PD[X[8, 1, 9, 2], X[12, 3, 13, 4], X[13, 20, 14, 21], X[9, 18, 10, 19], > X[21, 10, 22, 11], X[19, 14, 20, 7], X[5, 17, 6, 16], X[17, 22, 18, 15], > X[2, 7, 3, 8], X[4, 11, 5, 12], X[15, 1, 16, 6]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10, -7, 11}, {9, -1, -4, 5, 10, -2, -3, 6}, > {-11, 7, -8, 4, -6, 3, -5, 8}]
In[6]:=	Jones[L][q]
Out[6]=	-9 -8 -7 -5 -4 -2 1 1 + q - q + q + q + q + q - - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 2 3 3 4 3 4 2 2 -8 -2 1 + q + --- + --- + --- + --- + --- + --- + --- + --- + q + q 24 22 20 18 16 14 12 10 q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 437]][a, z]
Out[8]=	4 6 8 2 6 8 a 2 a a 2 2 4 2 8 2 2 4 2 a - 3 a + a + -- - ---- + -- + 4 a z - 5 a z + a z + a z - 2 2 2 z z z 4 4 4 6 > 5 a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 437]][a, z]
Out[9]=	4 6 8 5 7 2 4 6 8 a 2 a a 2 a 2 a 5 7 -2 a + 3 a + 11 a + 7 a - -- - ---- - -- + ---- + ---- - 9 a z - 9 a z + 2 2 2 z z z z z 2 2 4 2 6 2 8 2 10 2 3 3 5 3 > 6 a z + 3 a z - 25 a z - 17 a z + 5 a z + 5 a z + 13 a z + 7 3 9 3 2 4 4 4 6 4 8 4 10 4 > 13 a z + 5 a z - 5 a z - 5 a z + 21 a z + 16 a z - 5 a z - 3 5 5 5 7 5 9 5 2 6 4 6 6 6 8 6 > 5 a z - 7 a z - 7 a z - 5 a z + a z + a z - 8 a z - 7 a z + 10 6 3 7 5 7 7 7 9 7 6 8 8 8 > a z + a z + a z + a z + a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-5 2 1 1 1 1 1 1 1 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 19 8 17 8 17 7 15 6 13 6 15 5 13 5 q q t q t q t q t q t q t q t 1 1 3 1 1 1 2 2 4 > ------ + ------ + ------ + ----- + ------ + ----- + ----- + ----- + ----- + 11 5 13 4 11 4 9 4 11 3 9 3 7 3 9 2 7 2 q t q t q t q t q t q t q t q t q t 1 1 1 1 t 2 > ----- + ---- + ---- + ---- + -- + q t 5 2 7 5 3 3 q t q t q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n437

L11n436 L11n438