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

L11a188 L11a190

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a189 Visit L11a189's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X8192 X10,3,11,4 X16,6,17,5 X18,11,19,12 X22,15,7,16 X12,21,13,22 X20,13,21,14 X14,19,15,20 X4,18,5,17 X2738 X6,9,1,10

Gauss Code:

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

Jones Polynomial:

q-21/2 - 2q-19/2 + 5q-17/2 - 9q-15/2 + 12q-13/2 - 15q-11/2 + 15q-9/2 - 14q-7/2 + 10q-5/2 - 7q-3/2 + 3q-1/2 - q1/2

A2 (sl(3)) Invariant:

- q-32 - q-28 - 3q-26 + 2q-24 - q-22 + 4q-18 + 4q-14 + q-10 + 3q-8 - 2q-6 + 3q-4 - 1 + q2

HOMFLY-PT Polynomial:

- az - az3 - a3z-1 - 3a3z + a3z5 + 2a5z + 4a5z3 + 2a5z5 + 2a7z-1 + 3a7z + 2a7z3 + a7z5 - a9z-1 - 2a9z - a9z3

Kauffman Polynomial:

- az + 2az3 - az5 - a2z2 + 5a2z4 - 3a2z6 - a3z-1 + 5a3z - 6a3z3 + 9a3z5 - 5a3z7 + a4 - a4z2 - 2a4z4 + 7a4z6 - 5a4z8 + 3a5z - 13a5z3 + 11a5z5 - a5z7 - 3a5z9 - 3a6 + 9a6z2 - 20a6z4 + 17a6z6 - 6a6z8 - a6z10 + 2a7z-1 - 10a7z + 19a7z3 - 22a7z5 + 15a7z7 - 6a7z9 - 5a8 + 22a8z2 - 28a8z4 + 17a8z6 - 4a8z8 - a8z10 + a9z-1 - 7a9z + 20a9z3 - 17a9z5 + 9a9z7 - 3a9z9 - 2a10 + 9a10z2 - 11a10z4 + 9a10z6 - 3a10z8 - 4a11z3 + 6a11z5 - 2a11z7 - 4a12z2 + 4a12z4 - a12z6

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 2                       1 j = 0                     2   j = -2                   5 1   j = -4                 6 3     j = -6               8 4       j = -8             8 7         j = -10           7 7           j = -12         5 8             j = -14       4 7               j = -16     1 5                 j = -18   1 4                   j = -20   1                     j = -22 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, Alternating, 189]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 189]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 3, 11, 4], X[16, 6, 17, 5], X[18, 11, 19, 12], > X[22, 15, 7, 16], X[12, 21, 13, 22], X[20, 13, 21, 14], X[14, 19, 15, 20], > X[4, 18, 5, 17], X[2, 7, 3, 8], X[6, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -9, 3, -11}, > {10, -1, 11, -2, 4, -6, 7, -8, 5, -3, 9, -4, 8, -7, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 2 5 9 12 15 15 14 10 7 q - ----- + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q q 3 > ------- - Sqrt[q] Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 -28 3 2 -22 4 4 -10 3 2 3 2 -1 - q - q - --- + --- - q + --- + --- + q + -- - -- + -- + q 26 24 18 14 8 6 4 q q q q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 189]][a, z]
Out[8]=	3 7 9 a 2 a a 3 5 7 9 3 5 3 -(--) + ---- - -- - a z - 3 a z + 2 a z + 3 a z - 2 a z - a z + 4 a z + z z z 7 3 9 3 3 5 5 5 7 5 > 2 a z - a z + a z + 2 a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 189]][a, z]
Out[9]=	3 7 9 4 6 8 10 a 2 a a 3 5 7 a - 3 a - 5 a - 2 a - -- + ---- + -- - a z + 5 a z + 3 a z - 10 a z - z z z 9 2 2 4 2 6 2 8 2 10 2 12 2 > 7 a z - a z - a z + 9 a z + 22 a z + 9 a z - 4 a z + 3 3 3 5 3 7 3 9 3 11 3 2 4 > 2 a z - 6 a z - 13 a z + 19 a z + 20 a z - 4 a z + 5 a z - 4 4 6 4 8 4 10 4 12 4 5 3 5 > 2 a z - 20 a z - 28 a z - 11 a z + 4 a z - a z + 9 a z + 5 5 7 5 9 5 11 5 2 6 4 6 6 6 > 11 a z - 22 a z - 17 a z + 6 a z - 3 a z + 7 a z + 17 a z + 8 6 10 6 12 6 3 7 5 7 7 7 9 7 > 17 a z + 9 a z - a z - 5 a z - a z + 15 a z + 9 a z - 11 7 4 8 6 8 8 8 10 8 5 9 7 9 > 2 a z - 5 a z - 6 a z - 4 a z - 3 a z - 3 a z - 6 a z - 9 9 6 10 8 10 > 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 5 1 1 1 4 1 5 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 22 9 20 8 18 8 18 7 16 7 16 6 14 6 q q q t q t q t q t q t q t q t 7 5 8 7 7 8 7 8 4 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 14 5 12 5 12 4 10 4 10 3 8 3 8 2 6 2 6 q t q t q t q t q t q t q t q t q t 6 t 2 2 > ---- + 2 t + -- + q t 4 2 q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a189

L11a188 L11a190