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

L11a407 L11a409

Knotscape

This page is passe. Go here instead! The 3-Component Link L11a408 Visit L11a408's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-6 - 3q-5 + 7q-4 - 12q-3 + 17q-2 - 19q-1 + 21 - 17q + 14q2 - 8q3 + 4q4 - q5

A2 (sl(3)) Invariant:

q-20 + q-18 - 2q-16 + q-14 + q-12 - 4q-10 + 4q-8 + q-6 + 2q-4 + 7q-2 + 2 + 8q2 + 2q6 + 4q8 - 3q10 + 2q12 + q14 - q16

HOMFLY-PT Polynomial:

- a-4z2 + a-2z-2 + 2a-2 + 2a-2z2 + 2a-2z4 - 2z-2 - 4 - 5z2 - 2z4 - z6 + a2z-2 + 3a2 + 4a2z2 + 3a2z4 - 2a4 - 3a4z2 + a6

Kauffman Polynomial:

- a-5z3 + a-5z5 + 3a-4z2 - 6a-4z4 + 4a-4z6 + 4a-3z3 - 10a-3z5 + 7a-3z7 + a-2z-2 - 4a-2 + 6a-2z2 - 5a-2z4 - 5a-2z6 + 7a-2z8 - 2a-1z-1 + 4a-1z + 8a-1z3 - 20a-1z5 + 7a-1z7 + 4a-1z9 + 2z-2 - 6 + 7z2 + 5z4 - 22z6 + 13z8 + z10 - 2az-1 + 14az3 - 20az5 + 7az9 + a2z-2 - 3a2 + 6a2z2 + 5a2z4 - 20a2z6 + 10a2z8 + a2z10 - 6a3z + 18a3z3 - 19a3z5 + 3a3z7 + 3a3z9 - a4 + 5a4z2 - 2a4z4 - 6a4z6 + 4a4z8 - 2a5z + 7a5z3 - 8a5z5 + 3a5z7 - a6 + 3a6z2 - 3a6z4 + 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                   5 1   j = 5                 9 3     j = 3               10 7       j = 1             11 7         j = -1           9 11           j = -3         8 10             j = -5       4 9               j = -7     3 8                 j = -9   1 5                   j = -11   2                     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, 408]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 408]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[18, 10, 19, 9], X[14, 8, 15, 7], > X[20, 17, 21, 18], X[16, 21, 17, 22], X[22, 14, 11, 13], X[10, 16, 5, 15], > X[8, 20, 9, 19], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, -9, 3, -8}, > {11, -2, 7, -4, 8, -6, 5, -3, 9, -5, 6, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-6 3 7 12 17 19 2 3 4 5 21 + q - -- + -- - -- + -- - -- - 17 q + 14 q - 8 q + 4 q - q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -18 2 -14 -12 4 4 -6 2 7 2 6 2 + q + q - --- + q + q - --- + -- + q + -- + -- + 8 q + 2 q + 16 10 8 4 2 q q q q q 8 10 12 14 16 > 4 q - 3 q + 2 q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 408]][a, z]
Out[8]=	2 2 2 2 2 4 6 2 1 a 2 z 2 z 2 2 -4 + -- + 3 a - 2 a + a - -- + ----- + -- - 5 z - -- + ---- + 4 a z - 2 2 2 2 2 4 2 a z a z z a a 4 4 2 4 2 z 2 4 6 > 3 a z - 2 z + ---- + 3 a z - z 2 a
In[9]:=	Kauffman[Link[11, Alternating, 408]][a, z]
Out[9]=	2 4 2 4 6 2 1 a 2 2 a 4 z 3 -6 - -- - 3 a - a - a + -- + ----- + -- - --- - --- + --- - 6 a z - 2 2 2 2 2 a z z a a z a z z 2 2 3 3 5 2 3 z 6 z 2 2 4 2 6 2 z 4 z > 2 a z + 7 z + ---- + ---- + 6 a z + 5 a z + 3 a z - -- + ---- + 4 2 5 3 a a a a 3 4 4 8 z 3 3 3 5 3 4 6 z 5 z 2 4 > ---- + 14 a z + 18 a z + 7 a z + 5 z - ---- - ---- + 5 a z - a 4 2 a a 5 5 5 4 4 6 4 z 10 z 20 z 5 3 5 5 5 > 2 a z - 3 a z + -- - ----- - ----- - 20 a z - 19 a z - 8 a z - 5 3 a a a 6 6 7 7 6 4 z 5 z 2 6 4 6 6 6 7 z 7 z 3 7 > 22 z + ---- - ---- - 20 a z - 6 a z + a z + ---- + ---- + 3 a z + 4 2 3 a a a a 8 9 5 7 8 7 z 2 8 4 8 4 z 9 3 9 > 3 a z + 13 z + ---- + 10 a z + 4 a z + ---- + 7 a z + 3 a z + 2 a a 10 2 10 > z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	11 1 2 1 5 3 8 4 9 -- + 11 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 8 10 9 3 3 2 5 2 5 3 > ----- + ---- + --- + 7 q t + 10 q t + 7 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 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a408

L11a407 L11a409