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

L11a115 L11a117

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a116 Visit L11a116's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-25/2 - 2q-23/2 + 4q-21/2 - 6q-19/2 + 8q-17/2 - 10q-15/2 + 10q-13/2 - 10q-11/2 + 7q-9/2 - 6q-7/2 + 3q-5/2 - q-3/2

A2 (sl(3)) Invariant:

- q-40 - 2q-38 - q-34 - q-32 + 3q-30 + 2q-24 + 2q-20 + q-18 + 2q-16 + 3q-14 - q-12 + 2q-10 + q-8 - 2q-6 + q-4

HOMFLY-PT Polynomial:

- a3z3 - a5z-1 - 3a5z - 3a5z3 + a7z-1 - a7z - 3a7z3 - a9z-1 - a9z - 2a9z3 + 2a11z-1 + 3a11z - a13z-1

Kauffman Polynomial:

- a3z3 - 3a4z4 + a5z-1 - 3a5z + 6a5z3 - 6a5z5 - a6 + 8a6z4 - 7a6z6 + a7z-1 - 2a7z + 11a7z5 - 7a7z7 + a8z2 - 4a8z4 + 13a8z6 - 6a8z8 + a9z-1 - 5a9z + 4a9z3 - 4a9z5 + 10a9z7 - 4a9z9 - 4a10 + 23a10z2 - 48a10z4 + 34a10z6 - 5a10z8 - a10z10 + 2a11z-1 - 9a11z + 23a11z3 - 40a11z5 + 28a11z7 - 6a11z9 - 7a12 + 32a12z2 - 45a12z4 + 20a12z6 - a12z10 + a13z-1 - 3a13z + 12a13z3 - 19a13z5 + 11a13z7 - 2a13z9 - 3a14 + 10a14z2 - 12a14z4 + 6a14z6 - a14z8

Khovanov Homology:

trqj r = -11 r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -2                       1 j = -4                     3 1 j = -6                   3     j = -8                 4 3     j = -10               6 3       j = -12             5 5         j = -14           5 5           j = -16         3 5             j = -18       3 5               j = -20     1 3                 j = -22   1 3                   j = -24   1                     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, Alternating, 116]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 116]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[22, 15, 5, 16], X[16, 7, 17, 8], > X[8, 21, 9, 22], X[20, 9, 21, 10], X[18, 11, 19, 12], X[12, 17, 13, 18], > X[10, 19, 11, 20], X[2, 5, 3, 6], X[4, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, -5, 6, -9, 7, -8, 11, -2, 3, -4, 8, -7, > 9, -6, 5, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(25/2) 2 4 6 8 10 10 10 7 q - ----- + ----- - ----- + ----- - ----- + ----- - ----- + ---- - 23/2 21/2 19/2 17/2 15/2 13/2 11/2 9/2 q q q q q q q q 6 3 -(3/2) > ---- + ---- - q 7/2 5/2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-40 2 -34 -32 3 2 2 -18 2 3 -12 2 -q - --- - q - q + --- + --- + --- + q + --- + --- - q + --- + 38 30 24 20 16 14 10 q q q q q q q -8 2 -4 > q - -- + q 6 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 116]][a, z]
Out[8]=	5 7 9 11 13 a a a 2 a a 5 7 9 11 3 3 -(--) + -- - -- + ----- - --- - 3 a z - a z - a z + 3 a z - a z - z z z z z 5 3 7 3 9 3 > 3 a z - 3 a z - 2 a z
In[9]:=	Kauffman[Link[11, Alternating, 116]][a, z]
Out[9]=	5 7 9 11 13 6 10 12 14 a a a 2 a a 5 7 -a - 4 a - 7 a - 3 a + -- + -- + -- + ----- + --- - 3 a z - 2 a z - z z z z z 9 11 13 8 2 10 2 12 2 14 2 > 5 a z - 9 a z - 3 a z + a z + 23 a z + 32 a z + 10 a z - 3 3 5 3 9 3 11 3 13 3 4 4 6 4 > a z + 6 a z + 4 a z + 23 a z + 12 a z - 3 a z + 8 a z - 8 4 10 4 12 4 14 4 5 5 7 5 > 4 a z - 48 a z - 45 a z - 12 a z - 6 a z + 11 a z - 9 5 11 5 13 5 6 6 8 6 10 6 > 4 a z - 40 a z - 19 a z - 7 a z + 13 a z + 34 a z + 12 6 14 6 7 7 9 7 11 7 13 7 > 20 a z + 6 a z - 7 a z + 10 a z + 28 a z + 11 a z - 8 8 10 8 14 8 9 9 11 9 13 9 10 10 > 6 a z - 5 a z - a z - 4 a z - 6 a z - 2 a z - a z - 12 10 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 -2 1 1 1 3 1 3 3 q + q + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 26 11 24 10 22 10 22 9 20 9 20 8 18 8 q t q t q t q t q t q t q t 5 3 5 5 5 5 5 6 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 18 7 16 7 16 6 14 6 14 5 12 5 12 4 10 4 q t q t q t q t q t q t q t q t 3 4 3 3 3 > ------ + ----- + ----- + ----- + ---- 10 3 8 3 8 2 6 2 4 q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a116

L11a115 L11a117