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

L11n235 L11n237

Knotscape

This page is passe. Go here instead! The 2-Component Link L11n236 Visit L11n236's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X10,1,11,2 X2,11,3,12 X14,3,15,4 X20,13,21,14 X12,21,13,22 X22,5,9,6 X7,16,8,17 X15,18,16,19 X17,8,18,1 X6,9,7,10 X4,19,5,20

Gauss Code:

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

Jones Polynomial:

- q-25/2 + q-23/2 - q-21/2 + q-19/2 - 2q-11/2 + q-9/2 - q-7/2

A2 (sl(3)) Invariant:

- q-42 + q-40 + 2q-38 + q-34 - q-32 - q-30 - q-28 + 2q-24 + q-22 + 2q-20 + 2q-18 + q-16 + q-12

HOMFLY-PT Polynomial:

- a7z-1 - 6a7z - 10a7z3 - 6a7z5 - a7z7 + a9z-1 - 3a9z - 9a9z3 - 6a9z5 - a9z7 + 6a11z + 6a11z3 + a11z5 - a13z

Kauffman Polynomial:

- a7z-1 + 6a7z - 10a7z3 + 6a7z5 - a7z7 + a8 - 2a8z2 - 4a8z4 + 5a8z6 - a8z8 - a9z-1 - a9z + 4a9z3 - 9a9z5 + 6a9z7 - a9z9 + 10a10z2 - 22a10z4 + 13a10z6 - 2a10z8 - 7a11z + 15a11z3 - 15a11z5 + 7a11z7 - a11z9 + 10a12z2 - 14a12z4 + 7a12z6 - a12z8 - 3a13z + 5a13z3 - a13z5 - 2a14z2 + 4a14z4 - a14z6 - 3a15z + 4a15z3 - a15z5

Khovanov Homology:

trqj r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -6                     1 j = -8                   1 1 j = -10                 1     j = -12             2 1 1     j = -14           1 2 1       j = -16         1 2 1         j = -18       1 2 2           j = -20     1 1               j = -22   1 2 1               j = -24                       j = -26 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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 236]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 236]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[14, 3, 15, 4], X[20, 13, 21, 14], > X[12, 21, 13, 22], X[22, 5, 9, 6], X[7, 16, 8, 17], X[15, 18, 16, 19], > X[17, 8, 18, 1], X[6, 9, 7, 10], X[4, 19, 5, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -11, 6, -10, -7, 9}, > {10, -1, 2, -5, 4, -3, -8, 7, -9, 8, 11, -4, 5, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(25/2) -(23/2) -(21/2) -(19/2) 2 -(9/2) -(7/2) -q + q - q + q - ----- + q - q 11/2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 -40 2 -34 -32 -30 -28 2 -22 2 2 -q + q + --- + q - q - q - q + --- + q + --- + --- + 38 24 20 18 q q q q -16 -12 > q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 236]][a, z]
Out[8]=	7 9 a a 7 9 11 13 7 3 9 3 -(--) + -- - 6 a z - 3 a z + 6 a z - a z - 10 a z - 9 a z + z z 11 3 7 5 9 5 11 5 7 7 9 7 > 6 a z - 6 a z - 6 a z + a z - a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 236]][a, z]
Out[9]=	7 9 8 a a 7 9 11 13 15 8 2 a - -- - -- + 6 a z - a z - 7 a z - 3 a z - 3 a z - 2 a z + z z 10 2 12 2 14 2 7 3 9 3 11 3 > 10 a z + 10 a z - 2 a z - 10 a z + 4 a z + 15 a z + 13 3 15 3 8 4 10 4 12 4 14 4 > 5 a z + 4 a z - 4 a z - 22 a z - 14 a z + 4 a z + 7 5 9 5 11 5 13 5 15 5 8 6 10 6 > 6 a z - 9 a z - 15 a z - a z - a z + 5 a z + 13 a z + 12 6 14 6 7 7 9 7 11 7 8 8 10 8 > 7 a z - a z - a z + 6 a z + 7 a z - a z - 2 a z - 12 8 9 9 11 9 > a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-8 -6 1 1 2 1 1 1 1 q + q + ------- + ------ + ------ + ------ + ------ + ------ + ------ + 26 10 22 9 22 8 20 8 22 7 20 7 18 7 q t q t q t q t q t q t q t 2 1 2 2 1 1 2 2 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 18 6 16 6 18 5 16 5 14 5 16 4 14 4 12 4 q t q t q t q t q t q t q t q t 1 1 1 1 1 > ------ + ------ + ------ + ------ + ---- 14 3 12 3 12 2 10 2 8 q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n236

L11n235 L11n237