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

L11a492 L11a494

Knotscape

This page is passe. Go here instead! The 3-Component Link L11a493 Visit L11a493's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X12,6,13,5 X10,4,11,3 X18,12,5,11 X20,10,21,9 X22,16,19,15 X14,22,15,21 X2,14,3,13 X8,20,9,19

Gauss Code:

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

Jones Polynomial:

- q-1 + 4 - 8q + 15q2 - 18q3 + 24q4 - 23q5 + 20q6 - 16q7 + 10q8 - 4q9 + q10

A2 (sl(3)) Invariant:

- q-2 + 2 - 2q2 + 3q4 + 4q6 + q8 + 11q10 + 2q12 + 7q14 + q16 - 3q18 + 2q20 - 4q22 + 3q24 + q26 - q28 + q30

HOMFLY-PT Polynomial:

2a-8 + 2a-8z2 + a-8z4 + a-6z-2 - 5a-6 - 9a-6z2 - 7a-6z4 - 2a-6z6 - 2a-4z-2 + 2a-4 + 9a-4z2 + 10a-4z4 + 5a-4z6 + a-4z8 + a-2z-2 + a-2 - 2a-2z2 - 3a-2z4 - a-2z6

Kauffman Polynomial:

a-12z4 + 4a-11z5 - a-10 + 4a-10z2 - 8a-10z4 + 10a-10z6 - 4a-9z + 13a-9z3 - 23a-9z5 + 16a-9z7 + a-8 + 8a-8z4 - 24a-8z6 + 16a-8z8 - 13a-7z + 34a-7z3 - 39a-7z5 + 2a-7z7 + 9a-7z9 + a-6z-2 + 5a-6 - 25a-6z2 + 66a-6z4 - 76a-6z6 + 24a-6z8 + 2a-6z10 - 2a-5z-1 - 11a-5z + 32a-5z3 - 11a-5z5 - 26a-5z7 + 14a-5z9 + 2a-4z-2 + 4a-4 - 29a-4z2 + 66a-4z4 - 56a-4z6 + 12a-4z8 + 2a-4z10 - 2a-3z-1 - 3a-3z + 14a-3z3 - 2a-3z5 - 11a-3z7 + 5a-3z9 + a-2z-2 - 8a-2z2 + 17a-2z4 - 14a-2z6 + 4a-2z8 - a-1z + 3a-1z3 - 3a-1z5 + a-1z7

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 21                       1 j = 19                     3   j = 17                   7 1   j = 15                 9 3     j = 13               11 7       j = 11             14 11         j = 9           10 9           j = 7         8 14             j = 5       7 10               j = 3     3 10                 j = 1   1 5                   j = -1   3                     j = -3 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, 493]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 493]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[12, 6, 13, 5], > X[10, 4, 11, 3], X[18, 12, 5, 11], X[20, 10, 21, 9], X[22, 16, 19, 15], > X[14, 22, 15, 21], X[2, 14, 3, 13], X[8, 20, 9, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 5, -3}, {11, -7, 9, -8}, > {4, -1, 2, -11, 7, -5, 6, -4, 10, -9, 8, -2, 3, -6}]
In[6]:=	Jones[L][q]
Out[6]=	1 2 3 4 5 6 7 8 9 10 4 - - - 8 q + 15 q - 18 q + 24 q - 23 q + 20 q - 16 q + 10 q - 4 q + q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-2 2 4 6 8 10 12 14 16 18 2 - q - 2 q + 3 q + 4 q + q + 11 q + 2 q + 7 q + q - 3 q + 20 22 24 26 28 30 > 2 q - 4 q + 3 q + q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 493]][a, z]
Out[8]=	2 2 2 2 4 2 5 2 -2 1 2 1 2 z 9 z 9 z 2 z z -- - -- + -- + a + ----- - ----- + ----- + ---- - ---- + ---- - ---- + -- - 8 6 4 6 2 4 2 2 2 8 6 4 2 8 a a a a z a z a z a a a a a 4 4 4 6 6 6 8 7 z 10 z 3 z 2 z 5 z z z > ---- + ----- - ---- - ---- + ---- - -- + -- 6 4 2 6 4 2 4 a a a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 493]][a, z]
Out[9]=	-10 -8 5 4 1 2 1 2 2 4 z 13 z -a + a + -- + -- + ----- + ----- + ----- - ---- - ---- - --- - ---- - 6 4 6 2 4 2 2 2 5 3 9 7 a a a z a z a z a z a z a a 2 2 2 2 3 3 3 11 z 3 z z 4 z 25 z 29 z 8 z 13 z 34 z 32 z > ---- - --- - - + ---- - ----- - ----- - ---- + ----- + ----- + ----- + 5 3 a 10 6 4 2 9 7 5 a a a a a a a a a 3 3 4 4 4 4 4 4 5 5 14 z 3 z z 8 z 8 z 66 z 66 z 17 z 4 z 23 z > ----- + ---- + --- - ---- + ---- + ----- + ----- + ----- + ---- - ----- - 3 a 12 10 8 6 4 2 11 9 a a a a a a a a a 5 5 5 5 6 6 6 6 6 39 z 11 z 2 z 3 z 10 z 24 z 76 z 56 z 14 z > ----- - ----- - ---- - ---- + ----- - ----- - ----- - ----- - ----- + 7 5 3 a 10 8 6 4 2 a a a a a a a a 7 7 7 7 7 8 8 8 8 9 16 z 2 z 26 z 11 z z 16 z 24 z 12 z 4 z 9 z > ----- + ---- - ----- - ----- + -- + ----- + ----- + ----- + ---- + ---- + 9 7 5 3 a 8 6 4 2 7 a a a a a a a a a 9 9 10 10 14 z 5 z 2 z 2 z > ----- + ---- + ----- + ----- 5 3 6 4 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 5 1 3 q 5 q 3 q 5 7 7 2 10 q + 7 q + ----- + ---- + -- + --- + ---- + 10 q t + 8 q t + 14 q t + 3 3 2 2 t t q t q t t 9 2 9 3 11 3 11 4 13 4 13 5 > 10 q t + 9 q t + 14 q t + 11 q t + 11 q t + 7 q t + 15 5 15 6 17 6 17 7 19 7 21 8 > 9 q t + 3 q t + 7 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a493

L11a492 L11a494