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

L11a85 L11a87

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a86 Visit L11a86's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X14,10,15,9 X10,14,11,13 X20,17,21,18 X18,7,19,8 X8,19,9,20 X22,15,5,16 X16,21,17,22 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

q-21/2 - 2q-19/2 + 5q-17/2 - 9q-15/2 + 12q-13/2 - 15q-11/2 + 14q-9/2 - 14q-7/2 + 10q-5/2 - 6q-3/2 + 3q-1/2 - q1/2

A2 (sl(3)) Invariant:

- q-34 - 2q-32 - q-28 - 2q-26 + 4q-24 + q-22 + q-20 + 4q-18 + 4q-14 + q-10 + 2q-8 - 4q-6 + 2q-4 - 1 + q2

HOMFLY-PT Polynomial:

- az - az3 + a3z + a3z3 + a3z5 - a5z-1 - 3a5z - a5z3 + a5z5 - 2a7z - 3a7z3 + 2a9z-1 + 3a9z - a11z-1

Kauffman Polynomial:

- az + 2az3 - az5 - 3a2z2 + 6a2z4 - 3a2z6 + 5a3z5 - 4a3z7 - 3a4z2 + 3a4z4 + 3a4z6 - 4a4z8 + a5z-1 - 2a5z + 3a5z3 - 2a5z5 + 2a5z7 - 3a5z9 - a6 + 2a6z2 - 5a6z4 + 7a6z6 - 4a6z8 - a6z10 + 4a7z3 - 9a7z5 + 9a7z7 - 5a7z9 + 3a8 - 6a8z2 + 4a8z4 + 3a8z6 - 2a8z8 - a8z10 - 2a9z-1 + 5a9z - 6a9z3 + 5a9z5 + a9z7 - 2a9z9 + 5a10 - 13a10z2 + 10a10z4 + a10z6 - 2a10z8 - a11z-1 + 2a11z - 5a11z3 + 6a11z5 - 2a11z7 + 2a12 - 5a12z2 + 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                   4 1   j = -4                 7 3     j = -6               7 3       j = -8             7 7         j = -10           8 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, 86]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 86]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[14, 10, 15, 9], X[10, 14, 11, 13], > X[20, 17, 21, 18], X[18, 7, 19, 8], X[8, 19, 9, 20], X[22, 15, 5, 16], > X[16, 21, 17, 22], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 6, -7, 3, -4, 11, -2, 4, -3, 8, -9, 5, -6, > 7, -5, 9, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 2 5 9 12 15 14 14 10 6 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]=	-34 2 -28 2 4 -22 -20 4 4 -10 2 4 -1 - q - --- - q - --- + --- + q + q + --- + --- + q + -- - -- + 32 26 24 18 14 8 6 q q q q q q q 2 2 > -- + q 4 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 86]][a, z]
Out[8]=	5 9 11 a 2 a a 3 5 7 9 3 3 3 -(--) + ---- - --- - a z + a z - 3 a z - 2 a z + 3 a z - a z + a z - z z z 5 3 7 3 3 5 5 5 > a z - 3 a z + a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 86]][a, z]
Out[9]=	5 9 11 6 8 10 12 a 2 a a 5 9 -a + 3 a + 5 a + 2 a + -- - ---- - --- - a z - 2 a z + 5 a z + z z z 11 2 2 4 2 6 2 8 2 10 2 12 2 > 2 a z - 3 a z - 3 a z + 2 a z - 6 a z - 13 a z - 5 a z + 3 5 3 7 3 9 3 11 3 2 4 4 4 > 2 a z + 3 a z + 4 a z - 6 a z - 5 a z + 6 a z + 3 a z - 6 4 8 4 10 4 12 4 5 3 5 5 5 > 5 a z + 4 a z + 10 a z + 4 a z - a z + 5 a z - 2 a z - 7 5 9 5 11 5 2 6 4 6 6 6 8 6 > 9 a z + 5 a z + 6 a z - 3 a z + 3 a z + 7 a z + 3 a z + 10 6 12 6 3 7 5 7 7 7 9 7 11 7 > a z - a z - 4 a z + 2 a z + 9 a z + a z - 2 a z - 4 8 6 8 8 8 10 8 5 9 7 9 9 9 > 4 a z - 4 a z - 2 a z - 2 a z - 3 a z - 5 a z - 2 a z - 6 10 8 10 > a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 4 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 8 7 7 7 7 3 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 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 7 t 2 2 > ---- + 2 t + -- + q t 4 2 q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a86

L11a85 L11a87