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

L11a419 L11a421

Knotscape

This page is passe. Go here instead! The 3-Component Link L11a420 Visit L11a420's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-4 + 2q-3 - 5q-2 + 9q-1 - 11 + 15q - 14q2 + 14q3 - 10q4 + 7q5 - 3q6 + q7

A2 (sl(3)) Invariant:

- q-12 - q-10 - q-8 - 3q-6 + q-4 - q-2 + 3 + 5q2 + 2q4 + 8q6 + 2q8 + 6q10 + 3q12 + q14 + 3q16 - q18 + q20

HOMFLY-PT Polynomial:

2a-4z-2 + 5a-4 + 6a-4z2 + 4a-4z4 + a-4z6 - 5a-2z-2 - 16a-2 - 21a-2z2 - 15a-2z4 - 6a-2z6 - a-2z8 + 4z-2 + 15 + 18z2 + 10z4 + 2z6 - a2z-2 - 4a2 - 4a2z2 - a2z4

Kauffman Polynomial:

- a-8z2 + a-8z4 - 2a-7z3 + 3a-7z5 - 2a-6 + 6a-6z2 - 8a-6z4 + 6a-6z6 + a-5z + 2a-5z3 - 8a-5z5 + 7a-5z7 - 2a-4z-2 + 6a-4 - 9a-4z2 + 11a-4z4 - 12a-4z6 + 7a-4z8 + 5a-3z-1 - 16a-3z + 23a-3z3 - 15a-3z5 - a-3z7 + 4a-3z9 - 5a-2z-2 + 20a-2 - 39a-2z2 + 47a-2z4 - 35a-2z6 + 9a-2z8 + a-2z10 + 9a-1z-1 - 33a-1z + 39a-1z3 - 11a-1z5 - 12a-1z7 + 6a-1z9 - 4z-2 + 17 - 30z2 + 37z4 - 25z6 + 4z8 + z10 + 5az-1 - 21az + 28az3 - 12az5 - 3az7 + 2az9 - a2z-2 + 4a2 - 7a2z2 + 10a2z4 - 8a2z6 + 2a2z8 + a3z-1 - 5a3z + 8a3z3 - 5a3z5 + a3z7

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 15                       1 j = 13                     3 1 j = 11                   4     j = 9                 6 3     j = 7               8 4       j = 5             7 7         j = 3           8 7           j = 1         5 9             j = -1       4 6               j = -3     1 5                 j = -5   1 4                   j = -7   1                     j = -9 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, 420]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 420]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[20, 11, 21, 12], X[22, 15, 11, 16], > X[14, 21, 15, 22], X[8, 18, 9, 17], X[16, 8, 17, 7], X[18, 10, 19, 9], > X[10, 20, 5, 19], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 7, -6, 8, -9}, > {3, -2, 11, -5, 4, -7, 6, -8, 9, -3, 5, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-4 2 5 9 2 3 4 5 6 7 -11 - q + -- - -- + - + 15 q - 14 q + 14 q - 10 q + 7 q - 3 q + q 3 2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -10 -8 3 -4 -2 2 4 6 8 10 3 - q - q - q - -- + q - q + 5 q + 2 q + 8 q + 2 q + 6 q + 6 q 12 14 16 18 20 > 3 q + q + 3 q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 420]][a, z]
Out[8]=	2 2 2 5 16 2 4 2 5 a 2 6 z 21 z 15 + -- - -- - 4 a + -- + ----- - ----- - -- + 18 z + ---- - ----- - 4 2 2 4 2 2 2 2 4 2 a a z a z a z z a a 4 4 6 6 8 2 2 4 4 z 15 z 2 4 6 z 6 z z > 4 a z + 10 z + ---- - ----- - a z + 2 z + -- - ---- - -- 4 2 4 2 2 a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 420]][a, z]
Out[9]=	2 3 2 6 20 2 4 2 5 a 5 9 5 a a 17 - -- + -- + -- + 4 a - -- - ----- - ----- - -- + ---- + --- + --- + -- + 6 4 2 2 4 2 2 2 2 3 a z z z a a a z a z a z z a z 2 2 2 2 z 16 z 33 z 3 2 z 6 z 9 z 39 z > -- - ---- - ---- - 21 a z - 5 a z - 30 z - -- + ---- - ---- - ----- - 5 3 a 8 6 4 2 a a a a a a 3 3 3 3 4 2 2 2 z 2 z 23 z 39 z 3 3 3 4 z > 7 a z - ---- + ---- + ----- + ----- + 28 a z + 8 a z + 37 z + -- - 7 5 3 a 8 a a a a 4 4 4 5 5 5 5 8 z 11 z 47 z 2 4 3 z 8 z 15 z 11 z 5 > ---- + ----- + ----- + 10 a z + ---- - ---- - ----- - ----- - 12 a z - 6 4 2 7 5 3 a a a a a a a 6 6 6 7 7 7 3 5 6 6 z 12 z 35 z 2 6 7 z z 12 z > 5 a z - 25 z + ---- - ----- - ----- - 8 a z + ---- - -- - ----- - 6 4 2 5 3 a a a a a a 8 8 9 9 7 3 7 8 7 z 9 z 2 8 4 z 6 z 9 > 3 a z + a z + 4 z + ---- + ---- + 2 a z + ---- + ---- + 2 a z + 4 2 3 a a a a 10 10 z > z + --- 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 1 4 1 5 4 6 5 q 9 q + 8 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + --- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t q t q t 3 5 5 2 7 2 7 3 9 3 9 4 > 7 q t + 7 q t + 7 q t + 8 q t + 4 q t + 6 q t + 3 q t + 11 4 13 5 13 6 15 6 > 4 q t + 3 q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a420

L11a419 L11a421