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

L11n404 L11n406

Knotscape

This page is passe. Go here instead! The 3-Component Link L11n405 Visit L11n405's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X16,9,17,10 X8,15,9,16 X4,17,1,18 X22,12,19,11 X10,4,11,3 X5,21,6,20 X21,5,22,18 X12,20,13,19 X2,14,3,13

Gauss Code:

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

Jones Polynomial:

- q-5 + 3q-4 - 6q-3 + 8q-2 - 10q-1 + 12 - 9q + 8q2 - 4q3 + 3q4

A2 (sl(3)) Invariant:

- q-14 + q-12 - 2q-10 - 2q-6 - 3q-4 + 2q-2 - 1 + 6q2 + 3q4 + 6q6 + 7q8 + 4q10 + 5q12 + q14 + q16

HOMFLY-PT Polynomial:

2a-4z-2 + 3a-4 + a-4z2 - 5a-2z-2 - 11a-2 - 10a-2z2 - 5a-2z4 - a-2z6 + 4z-2 + 11 + 14z2 + 13z4 + 6z6 + z8 - a2z-2 - 3a2 - 5a2z2 - 4a2z4 - a2z6

Kauffman Polynomial:

- 2a-4z-2 + 10a-4 - 17a-4z2 + 6a-4z4 + 5a-3z-1 - 15a-3z + 14a-3z3 - 9a-3z5 + 3a-3z7 - 5a-2z-2 + 22a-2 - 45a-2z2 + 46a-2z4 - 22a-2z6 + 5a-2z8 + 9a-1z-1 - 29a-1z + 34a-1z3 - 12a-1z5 - a-1z7 + 2a-1z9 - 4z-2 + 17 - 38z2 + 56z4 - 34z6 + 9z8 + 5az-1 - 17az + 23az3 - 11az5 + 2az9 - a2z-2 + 4a2 - 9a2z2 + 10a2z4 - 9a2z6 + 4a2z8 + a3z-1 - 2a3z + a3z3 - 7a3z5 + 4a3z7 + a4z2 - 6a4z4 + 3a4z6 + a5z - 2a5z3 + a5z5

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 9                   3 j = 7                 3 2 j = 5               5 1   j = 3             4 3     j = 1           8 5       j = -1         4 6         j = -3       4 6           j = -5     2 4             j = -7   1 4               j = -9   2                 j = -11 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, NonAlternating, 405]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 405]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[16, 9, 17, 10], X[8, 15, 9, 16], > X[4, 17, 1, 18], X[22, 12, 19, 11], X[10, 4, 11, 3], X[5, 21, 6, 20], > X[21, 5, 22, 18], X[12, 20, 13, 19], X[2, 14, 3, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 7, -5}, {10, 8, -9, -6}, > {-8, -1, 2, -4, 3, -7, 6, -10, 11, -2, 4, -3, 5, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-5 3 6 8 10 2 3 4 12 - q + -- - -- + -- - -- - 9 q + 8 q - 4 q + 3 q 4 3 2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -12 2 2 3 2 2 4 6 8 10 -1 - q + q - --- - -- - -- + -- + 6 q + 3 q + 6 q + 7 q + 4 q + 10 6 4 2 q q q q 12 14 16 > 5 q + q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 405]][a, z]
Out[8]=	2 2 2 3 11 2 4 2 5 a 2 z 10 z 2 2 11 + -- - -- - 3 a + -- + ----- - ----- - -- + 14 z + -- - ----- - 5 a z + 4 2 2 4 2 2 2 2 4 2 a a z a z a z z a a 4 6 4 5 z 2 4 6 z 2 6 8 > 13 z - ---- - 4 a z + 6 z - -- - a z + z 2 2 a a
In[9]:=	Kauffman[Link[11, NonAlternating, 405]][a, z]
Out[9]=	2 3 10 22 2 4 2 5 a 5 9 5 a a 15 z 17 + -- + -- + 4 a - -- - ----- - ----- - -- + ---- + --- + --- + -- - ---- - 4 2 2 4 2 2 2 2 3 a z z z 3 a a z a z a z z a z a 2 2 29 z 3 5 2 17 z 45 z 2 2 4 2 > ---- - 17 a z - 2 a z + a z - 38 z - ----- - ----- - 9 a z + a z + a 4 2 a a 3 3 4 4 14 z 34 z 3 3 3 5 3 4 6 z 46 z > ----- + ----- + 23 a z + a z - 2 a z + 56 z + ---- + ----- + 3 a 4 2 a a a 5 5 2 4 4 4 9 z 12 z 5 3 5 5 5 6 > 10 a z - 6 a z - ---- - ----- - 11 a z - 7 a z + a z - 34 z - 3 a a 6 7 7 8 22 z 2 6 4 6 3 z z 3 7 8 5 z 2 8 > ----- - 9 a z + 3 a z + ---- - -- + 4 a z + 9 z + ---- + 4 a z + 2 3 a 2 a a a 9 2 z 9 > ---- + 2 a z a
In[10]:=	Kh[L][q, t]
Out[10]=	6 1 2 1 4 2 4 4 6 4 - + 8 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 7 4 9 4 > 5 q t + 4 q t + 3 q t + 5 q t + q t + 3 q t + 2 q t + 3 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n405

L11n404 L11n406