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

L11a518 L11a520

Knotscape

This page is passe. Go here instead! The 3-Component Link L11a519 Visit L11a519's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X8192 X16,6,17,5 X22,20,13,19 X18,10,19,9 X10,21,11,22 X20,11,21,12 X2,13,3,14 X14,3,15,4 X4758 X12,16,7,15 X6,18,1,17

Gauss Code:

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

Jones Polynomial:

q-6 - 4q-5 + 9q-4 - 14q-3 + 21q-2 - 22q-1 + 24 - 20q + 15q2 - 9q3 + 4q4 - q5

A2 (sl(3)) Invariant:

q-18 - q-16 + 3q-12 - q-10 + 8q-8 + 4q-6 + 4q-4 + 7q-2 - 2 + 6q2 - 3q4 + q6 + 2q8 - 3q10 + 2q12 - q14

HOMFLY-PT Polynomial:

- a-2 - 3a-2z2 - 3a-2z4 - a-2z6 + z-2 + 4 + 9z2 + 10z4 + 5z6 + z8 - 2a2z-2 - 4a2 - 8a2z2 - 7a2z4 - 2a2z6 + a4z-2 + a4 + 2a4z2 + a4z4

Kauffman Polynomial:

- a-5z3 + a-5z5 + a-4z2 - 5a-4z4 + 4a-4z6 - a-3z + 6a-3z3 - 12a-3z5 + 8a-3z7 + 2a-2 - 6a-2z2 + 12a-2z4 - 16a-2z6 + 10a-2z8 - 3a-1z + 12a-1z3 - 12a-1z5 - 2a-1z7 + 7a-1z9 - z-2 + 7 - 25z2 + 48z4 - 47z6 + 17z8 + 2z10 + 2az-1 - 4az + 6az3 + 3az5 - 20az7 + 13az9 - 2a2z-2 + 7a2 - 25a2z2 + 45a2z4 - 44a2z6 + 14a2z8 + 2a2z10 + 2a3z-1 - 2a3z + 6a3z3 - 7a3z5 - 6a3z7 + 6a3z9 - a4z-2 + 3a4 - 6a4z2 + 12a4z4 - 16a4z6 + 7a4z8 + 5a5z3 - 9a5z5 + 4a5z7 + a6z2 - 2a6z4 + a6z6

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 11                       1 j = 9                     3   j = 7                   6 1   j = 5                 9 3     j = 3               11 6       j = 1             13 9         j = -1           11 13           j = -3         10 11             j = -5       6 13               j = -7     3 8                 j = -9   1 6                   j = -11   3                     j = -13 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, 519]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 519]]
Out[4]=	PD[X[8, 1, 9, 2], X[16, 6, 17, 5], X[22, 20, 13, 19], X[18, 10, 19, 9], > X[10, 21, 11, 22], X[20, 11, 21, 12], X[2, 13, 3, 14], X[14, 3, 15, 4], > X[4, 7, 5, 8], X[12, 16, 7, 15], X[6, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -7, 8, -9, 2, -11}, {9, -1, 4, -5, 6, -10}, > {7, -8, 10, -2, 11, -4, 3, -6, 5, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-6 4 9 14 21 22 2 3 4 5 24 + q - -- + -- - -- + -- - -- - 20 q + 15 q - 9 q + 4 q - q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 3 -10 8 4 4 7 2 4 6 8 -2 + q - q + --- - q + -- + -- + -- + -- + 6 q - 3 q + q + 2 q - 12 8 6 4 2 q q q q q 10 12 14 > 3 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 519]][a, z]
Out[8]=	2 4 2 -2 2 4 -2 2 a a 2 3 z 2 2 4 2 4 - a - 4 a + a + z - ---- + -- + 9 z - ---- - 8 a z + 2 a z + 2 2 2 z z a 4 6 4 3 z 2 4 4 4 6 z 2 6 8 > 10 z - ---- - 7 a z + a z + 5 z - -- - 2 a z + z 2 2 a a
In[9]:=	Kauffman[Link[11, Alternating, 519]][a, z]
Out[9]=	2 4 3 2 2 4 -2 2 a a 2 a 2 a z 3 z 7 + -- + 7 a + 3 a - z - ---- - -- + --- + ---- - -- - --- - 4 a z - 2 2 2 z z 3 a a z z a 2 2 3 3 3 2 z 6 z 2 2 4 2 6 2 z 6 z > 2 a z - 25 z + -- - ---- - 25 a z - 6 a z + a z - -- + ---- + 4 2 5 3 a a a a 3 4 4 12 z 3 3 3 5 3 4 5 z 12 z 2 4 > ----- + 6 a z + 6 a z + 5 a z + 48 z - ---- + ----- + 45 a z + a 4 2 a a 5 5 5 4 4 6 4 z 12 z 12 z 5 3 5 5 5 > 12 a z - 2 a z + -- - ----- - ----- + 3 a z - 7 a z - 9 a z - 5 3 a a a 6 6 7 7 6 4 z 16 z 2 6 4 6 6 6 8 z 2 z > 47 z + ---- - ----- - 44 a z - 16 a z + a z + ---- - ---- - 4 2 3 a a a a 8 9 7 3 7 5 7 8 10 z 2 8 4 8 7 z > 20 a z - 6 a z + 4 a z + 17 z + ----- + 14 a z + 7 a z + ---- + 2 a a 9 3 9 10 2 10 > 13 a z + 6 a z + 2 z + 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	13 1 3 1 6 3 8 6 13 -- + 13 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 10 11 11 3 3 2 5 2 5 3 > ----- + ---- + --- + 9 q t + 11 q t + 6 q t + 9 q t + 3 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a519

L11a518 L11a520