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

L11a487 L11a489

Knotscape

This page is passe. Go here instead! The 3-Component Link L11a488 Visit L11a488's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

1 - 3q + 8q2 - 12q3 + 19q4 - 20q5 + 21q6 - 18q7 + 14q8 - 8q9 + 3q10 - q11

A2 (sl(3)) Invariant:

1 - q2 + q4 + 3q6 - 2q8 + 7q10 + 3q12 + 5q14 + 8q16 + q18 + 7q20 - q22 + q24 + q26 - 5q28 - q32 - q34

HOMFLY-PT Polynomial:

- a-10z-2 - 2a-10 - a-10z2 + 4a-8z-2 + 9a-8 + 8a-8z2 + 3a-8z4 - 5a-6z-2 - 13a-6 - 12a-6z2 - 7a-6z4 - 2a-6z6 + 2a-4z-2 + 5a-4 + 3a-4z2 - a-4z4 - a-4z6 + a-2 + 2a-2z2 + a-2z4

Kauffman Polynomial:

a-13z - 2a-13z3 + a-13z5 + a-12z2 - 4a-12z4 + 3a-12z6 + a-11z-1 - 4a-11z + 7a-11z3 - 9a-11z5 + 6a-11z7 - a-10z-2 + 5a-10 - 13a-10z2 + 16a-10z4 - 14a-10z6 + 8a-10z8 + 5a-9z-1 - 23a-9z + 39a-9z3 - 28a-9z5 + 4a-9z7 + 5a-9z9 - 4a-8z-2 + 19a-8 - 48a-8z2 + 71a-8z4 - 56a-8z6 + 19a-8z8 + a-8z10 + 9a-7z-1 - 35a-7z + 50a-7z3 - 29a-7z5 - 5a-7z7 + 9a-7z9 - 5a-6z-2 + 22a-6 - 48a-6z2 + 64a-6z4 - 51a-6z6 + 16a-6z8 + a-6z10 + 5a-5z-1 - 17a-5z + 24a-5z3 - 18a-5z5 + 4a-5z9 - 2a-4z-2 + 8a-4 - 11a-4z2 + 10a-4z4 - 11a-4z6 + 5a-4z8 + 4a-3z3 - 7a-3z5 + 3a-3z7 - a-2 + 3a-2z2 - 3a-2z4 + a-2z6

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 23                       1 j = 21                     2   j = 19                   6 1   j = 17                 8 2     j = 15               10 6       j = 13             11 8         j = 11           11 12           j = 9         8 9             j = 7       4 11               j = 5     4 8                 j = 3   1 6                   j = 1   2                     j = -1 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, Alternating, 488]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 488]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[12, 8, 13, 7], X[16, 10, 17, 9], > X[22, 18, 19, 17], X[20, 14, 21, 13], X[14, 20, 15, 19], X[18, 22, 5, 21], > X[8, 16, 9, 15], X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {7, -6, 8, -5}, > {10, -1, 3, -9, 4, -2, 11, -3, 6, -7, 9, -4, 5, -8}]
In[6]:=	Jones[L][q]
Out[6]=	2 3 4 5 6 7 8 9 10 1 - 3 q + 8 q - 12 q + 19 q - 20 q + 21 q - 18 q + 14 q - 8 q + 3 q - 11 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 4 6 8 10 12 14 16 18 20 22 1 - q + q + 3 q - 2 q + 7 q + 3 q + 5 q + 8 q + q + 7 q - q + 24 26 28 32 34 > q + q - 5 q - q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 488]][a, z]
Out[8]=	2 2 -2 9 13 5 -2 1 4 5 2 z 8 z --- + -- - -- + -- + a - ------ + ----- - ----- + ----- - --- + ---- - 10 8 6 4 10 2 8 2 6 2 4 2 10 8 a a a a a z a z a z a z a a 2 2 2 4 4 4 4 6 6 12 z 3 z 2 z 3 z 7 z z z 2 z z > ----- + ---- + ---- + ---- - ---- - -- + -- - ---- - -- 6 4 2 8 6 4 2 6 4 a a a a a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 488]][a, z]
Out[9]=	5 19 22 8 -2 1 4 5 2 1 5 --- + -- + -- + -- - a - ------ - ----- - ----- - ----- + ----- + ---- + 10 8 6 4 10 2 8 2 6 2 4 2 11 9 a a a a a z a z a z a z a z a z 2 2 2 9 5 z 4 z 23 z 35 z 17 z z 13 z 48 z > ---- + ---- + --- - --- - ---- - ---- - ---- + --- - ----- - ----- - 7 5 13 11 9 7 5 12 10 8 a z a z a a a a a a a a 2 2 2 3 3 3 3 3 3 4 48 z 11 z 3 z 2 z 7 z 39 z 50 z 24 z 4 z 4 z > ----- - ----- + ---- - ---- + ---- + ----- + ----- + ----- + ---- - ---- + 6 4 2 13 11 9 7 5 3 12 a a a a a a a a a a 4 4 4 4 4 5 5 5 5 5 16 z 71 z 64 z 10 z 3 z z 9 z 28 z 29 z 18 z > ----- + ----- + ----- + ----- - ---- + --- - ---- - ----- - ----- - ----- - 10 8 6 4 2 13 11 9 7 5 a a a a a a a a a a 5 6 6 6 6 6 6 7 7 7 7 z 3 z 14 z 56 z 51 z 11 z z 6 z 4 z 5 z > ---- + ---- - ----- - ----- - ----- - ----- + -- + ---- + ---- - ---- + 3 12 10 8 6 4 2 11 9 7 a a a a a a a a a a 7 8 8 8 8 9 9 9 10 10 3 z 8 z 19 z 16 z 5 z 5 z 9 z 4 z z z > ---- + ---- + ----- + ----- + ---- + ---- + ---- + ---- + --- + --- 3 10 8 6 4 9 7 5 8 6 a a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 5 1 2 q q 5 7 7 2 9 2 6 q + 4 q + ---- + --- + -- + 8 q t + 4 q t + 11 q t + 8 q t + 2 t t q t 9 3 11 3 11 4 13 4 13 5 15 5 > 9 q t + 11 q t + 12 q t + 11 q t + 8 q t + 10 q t + 15 6 17 6 17 7 19 7 19 8 21 8 23 9 > 6 q t + 8 q t + 2 q t + 6 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a488

L11a487 L11a489