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

L11a251 L11a253

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a252 Visit L11a252's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X10,1,11,2 X20,13,21,14 X12,4,13,3 X2,19,3,20 X14,5,15,6 X16,7,17,8 X8,9,1,10 X18,12,19,11 X6,15,7,16 X22,18,9,17 X4,22,5,21

Gauss Code:

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

Jones Polynomial:

q-17/2 - 4q-15/2 + 8q-13/2 - 13q-11/2 + 17q-9/2 - 20q-7/2 + 19q-5/2 - 17q-3/2 + 12q-1/2 - 8q1/2 + 4q3/2 - q5/2

A2 (sl(3)) Invariant:

- q-24 + 2q-22 - 2q-20 + 3q-18 + 4q-12 - 3q-10 + 6q-8 - 2q-6 + 2q-4 + q-2 - 2 + 2q2 - 2q4 + q6

HOMFLY-PT Polynomial:

- az - 4az3 - 4az5 - az7 - a3z-1 - a3z + 8a3z3 + 12a3z5 + 6a3z7 + a3z9 + a5z-1 - 4a5z3 - 4a5z5 - a5z7

Kauffman Polynomial:

- 2a-1z3 + 3a-1z5 - a-1z7 + 3z2 - 13z4 + 14z6 - 4z8 - az + 7az3 - 21az5 + 21az7 - 6az9 + 7a2z2 - 28a2z4 + 24a2z6 - 3a2z10 - a3z-1 + 2a3z + 10a3z3 - 38a3z5 + 43a3z7 - 14a3z9 + a4 + 5a4z2 - 28a4z4 + 34a4z6 - 7a4z8 - 3a4z10 - a5z-1 + 3a5z - 6a5z3 + 4a5z5 + 10a5z7 - 8a5z9 - 5a6z4 + 16a6z6 - 11a6z8 - 5a7z3 + 14a7z5 - 11a7z7 - a8z2 + 7a8z4 - 8a8z6 + 2a9z3 - 4a9z5 - a10z4

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 6                       1 j = 4                     3   j = 2                   5 1   j = 0                 7 3     j = -2               10 5       j = -4             10 8         j = -6           10 9           j = -8         7 10             j = -10       6 10               j = -12     3 8                 j = -14   1 5                   j = -16   3                     j = -18 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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 252]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 252]]
Out[4]=	PD[X[10, 1, 11, 2], X[20, 13, 21, 14], X[12, 4, 13, 3], X[2, 19, 3, 20], > X[14, 5, 15, 6], X[16, 7, 17, 8], X[8, 9, 1, 10], X[18, 12, 19, 11], > X[6, 15, 7, 16], X[22, 18, 9, 17], X[4, 22, 5, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -11, 5, -9, 6, -7}, > {7, -1, 8, -3, 2, -5, 9, -6, 10, -8, 4, -2, 11, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 4 8 13 17 20 19 17 12 q - ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q 3/2 5/2 > 8 Sqrt[q] + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 2 2 3 4 3 6 2 2 -2 2 4 6 -2 - q + --- - --- + --- + --- - --- + -- - -- + -- + q + 2 q - 2 q + q 22 20 18 12 10 8 6 4 q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 252]][a, z]
Out[8]=	3 5 a a 3 3 3 3 5 3 5 3 5 -(--) + -- - a z - a z - 4 a z + 8 a z - 4 a z - 4 a z + 12 a z - z z 5 5 7 3 7 5 7 3 9 > 4 a z - a z + 6 a z - a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 252]][a, z]
Out[9]=	3 5 4 a a 3 5 2 2 2 4 2 8 2 a - -- - -- - a z + 2 a z + 3 a z + 3 z + 7 a z + 5 a z - a z - z z 3 2 z 3 3 3 5 3 7 3 9 3 4 2 4 > ---- + 7 a z + 10 a z - 6 a z - 5 a z + 2 a z - 13 z - 28 a z - a 5 4 4 6 4 8 4 10 4 3 z 5 3 5 > 28 a z - 5 a z + 7 a z - a z + ---- - 21 a z - 38 a z + a 5 5 7 5 9 5 6 2 6 4 6 6 6 > 4 a z + 14 a z - 4 a z + 14 z + 24 a z + 34 a z + 16 a z - 7 8 6 z 7 3 7 5 7 7 7 8 4 8 > 8 a z - -- + 21 a z + 43 a z + 10 a z - 11 a z - 4 z - 7 a z - a 6 8 9 3 9 5 9 2 10 4 10 > 11 a z - 6 a z - 14 a z - 8 a z - 3 a z - 3 a z
In[10]:=	Kh[L][q, t]
Out[10]=	8 10 1 3 1 5 3 8 6 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 18 7 16 6 14 6 14 5 12 5 12 4 10 4 q q q t q t q t q t q t q t q t 10 7 10 10 9 10 5 t 2 2 2 > ------ + ----- + ----- + ----- + ---- + ---- + 7 t + --- + 3 t + 5 q t + 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q 2 3 4 3 6 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a252

L11a251 L11a253