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

L11a505 L11a507

Knotscape

This page is passe. Go here instead! The 3-Component Link L11a506 Visit L11a506's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X8192 X16,8,17,7 X14,6,15,5 X10,4,11,3 X4,14,5,13 X2,18,3,17 X18,9,19,10 X12,21,7,22 X22,11,13,12 X20,16,21,15 X6,19,1,20

Gauss Code:

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

Jones Polynomial:

- q-4 + 5q-3 - 10q-2 + 17q-1 - 21 + 26q - 24q2 + 22q3 - 16q4 + 9q5 - 4q6 + q7

A2 (sl(3)) Invariant:

- q-12 + 2q-10 + q-8 + 7q-4 + 6 + 5q2 + q4 + 8q6 - 3q8 + 4q10 - 2q12 - 3q14 + 3q16 - 2q18 + q20

HOMFLY-PT Polynomial:

3a-4z2 + 3a-4z4 + a-4z6 + a-2z-2 + a-2 - 7a-2z2 - 10a-2z4 - 5a-2z6 - a-2z8 - 2z-2 - 2 + 4z2 + 6z4 + 2z6 + a2z-2 + a2 - a2z2 - a2z4

Kauffman Polynomial:

a-8z4 - a-7z3 + 4a-7z5 + a-6z2 - 6a-6z4 + 9a-6z6 + 9a-5z3 - 20a-5z5 + 15a-5z7 - 8a-4z2 + 24a-4z4 - 34a-4z6 + 18a-4z8 + 10a-3z3 - 8a-3z5 - 17a-3z7 + 13a-3z9 + a-2z-2 - 2a-2 - 21a-2z2 + 71a-2z4 - 77a-2z6 + 19a-2z8 + 4a-2z10 - 2a-1z-1 + 2a-1z - 2a-1z3 + 34a-1z5 - 56a-1z7 + 21a-1z9 + 2z-2 - 3 - 14z2 + 53z4 - 49z6 + 6z8 + 4z10 - 2az-1 + 2az - az3 + 16az5 - 23az7 + 8az9 + a2z-2 - 2a2 - 2a2z2 + 13a2z4 - 15a2z6 + 5a2z8 + a3z3 - 2a3z5 + 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   j = 11                   6 1   j = 9                 10 3     j = 7               12 6       j = 5             13 11         j = 3           13 11           j = 1         10 15             j = -1       7 11               j = -3     4 11                 j = -5   1 6                   j = -7   4                     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, 506]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 506]]
Out[4]=	PD[X[8, 1, 9, 2], X[16, 8, 17, 7], X[14, 6, 15, 5], X[10, 4, 11, 3], > X[4, 14, 5, 13], X[2, 18, 3, 17], X[18, 9, 19, 10], X[12, 21, 7, 22], > X[22, 11, 13, 12], X[20, 16, 21, 15], X[6, 19, 1, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -6, 4, -5, 3, -11}, {2, -1, 7, -4, 9, -8}, > {5, -3, 10, -2, 6, -7, 11, -10, 8, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-4 5 10 17 2 3 4 5 6 7 -21 - q + -- - -- + -- + 26 q - 24 q + 22 q - 16 q + 9 q - 4 q + q 3 2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 2 -8 7 2 4 6 8 10 12 14 6 - q + --- + q + -- + 5 q + q + 8 q - 3 q + 4 q - 2 q - 3 q + 10 4 q q 16 18 20 > 3 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 506]][a, z]
Out[8]=	2 2 2 4 -2 2 2 1 a 2 3 z 7 z 2 2 4 3 z -2 + a + a - -- + ----- + -- + 4 z + ---- - ---- - a z + 6 z + ---- - 2 2 2 2 4 2 4 z a z z a a a 4 6 6 8 10 z 2 4 6 z 5 z z > ----- - a z + 2 z + -- - ---- - -- 2 4 2 2 a a a a
In[9]:=	Kauffman[Link[11, Alternating, 506]][a, z]
Out[9]=	2 2 2 2 2 1 a 2 2 a 2 z 2 z -3 - -- - 2 a + -- + ----- + -- - --- - --- + --- + 2 a z - 14 z + -- - 2 2 2 2 2 a z z a 6 a z a z z a 2 2 3 3 3 3 8 z 21 z 2 2 z 9 z 10 z 2 z 3 3 3 4 > ---- - ----- - 2 a z - -- + ---- + ----- - ---- - a z + a z + 53 z + 4 2 7 5 3 a a a a a a 4 4 4 4 5 5 5 5 z 6 z 24 z 71 z 2 4 4 z 20 z 8 z 34 z > -- - ---- + ----- + ----- + 13 a z + ---- - ----- - ---- + ----- + 8 6 4 2 7 5 3 a a a a a a a a 6 6 6 7 5 3 5 6 9 z 34 z 77 z 2 6 15 z > 16 a z - 2 a z - 49 z + ---- - ----- - ----- - 15 a z + ----- - 6 4 2 5 a a a a 7 7 8 8 9 17 z 56 z 7 3 7 8 18 z 19 z 2 8 13 z > ----- - ----- - 23 a z + a z + 6 z + ----- + ----- + 5 a z + ----- + 3 a 4 2 3 a a a a 9 10 21 z 9 10 4 z > ----- + 8 a z + 4 z + ----- a 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 4 1 6 4 11 7 11 15 q + 13 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t q t q t q t q t q t q t q t 10 q 3 5 5 2 7 2 7 3 9 3 > ---- + 11 q t + 13 q t + 11 q t + 12 q t + 6 q t + 10 q t + t 9 4 11 4 11 5 13 5 15 6 > 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a506

L11a505 L11a507