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

L11a333 L11a335

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a334 Visit L11a334's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 3q-11/2 - 8q-9/2 + 14q-7/2 - 20q-5/2 + 22q-3/2 - 23q-1/2 + 20q1/2 - 15q3/2 + 9q5/2 - 4q7/2 + q9/2

A2 (sl(3)) Invariant:

q-20 + 4q-14 - 3q-12 + 3q-10 + 3q-8 - 2q-6 + 5q-4 - 4q-2 + 3 - q2 - 2q4 + 4q6 - 3q8 + q10 + q12 - q14

HOMFLY-PT Polynomial:

a-3z + a-3z3 - 2a-1z - 4a-1z3 - 2a-1z5 + 2az + 4az3 + 3az5 + az7 - a3z-1 - 5a3z - 5a3z3 - 2a3z5 + a5z-1 + 2a5z + a5z3

Kauffman Polynomial:

- a-4z2 + 2a-4z4 - a-4z6 + a-3z - 6a-3z3 + 9a-3z5 - 4a-3z7 + 2a-2z2 - 9a-2z4 + 15a-2z6 - 7a-2z8 + 2a-1z - 8a-1z3 + 10a-1z5 + 5a-1z7 - 6a-1z9 + 6z2 - 24z4 + 36z6 - 13z8 - 2z10 + 4az - 13az3 + 7az5 + 15az7 - 12az9 + 6a2z2 - 27a2z4 + 35a2z6 - 14a2z8 - 2a2z10 - a3z-1 + 9a3z - 21a3z3 + 16a3z5 - 6a3z9 + a4 + 2a4z2 - 10a4z4 + 12a4z6 - 8a4z8 - a5z-1 + 5a5z - 8a5z3 + 9a5z5 - 6a5z7 - a6z2 + 4a6z4 - 3a6z6 - a7z + 2a7z3 - a7z5

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 = 10                       1 j = 8                     3   j = 6                   6 1   j = 4                 9 3     j = 2               11 6       j = 0             12 9         j = -2           11 12           j = -4         9 11             j = -6       5 11               j = -8     3 9                 j = -10   1 6                   j = -12   2                     j = -14 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, 334]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 334]]
Out[4]=	PD[X[10, 1, 11, 2], X[20, 6, 21, 5], X[14, 3, 15, 4], X[4, 15, 5, 16], > X[22, 20, 9, 19], X[16, 7, 17, 8], X[18, 12, 19, 11], X[12, 18, 13, 17], > X[2, 9, 3, 10], X[8, 13, 1, 14], X[6, 22, 7, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 3, -4, 2, -11, 6, -10}, > {9, -1, 7, -8, 10, -3, 4, -6, 8, -7, 5, -2, 11, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 3 8 14 20 22 23 -q + ----- - ---- + ---- - ---- + ---- - ------- + 20 Sqrt[q] - 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 3/2 5/2 7/2 9/2 > 15 q + 9 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 4 3 3 3 2 5 4 2 4 6 8 3 + q + --- - --- + --- + -- - -- + -- - -- - q - 2 q + 4 q - 3 q + 14 12 10 8 6 4 2 q q q q q q q 10 12 14 > q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 334]][a, z]
Out[8]=	3 5 3 3 a a z 2 z 3 5 z 4 z 3 -(--) + -- + -- - --- + 2 a z - 5 a z + 2 a z + -- - ---- + 4 a z - z z 3 a 3 a a a 5 3 3 5 3 2 z 5 3 5 7 > 5 a z + a z - ---- + 3 a z - 2 a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 334]][a, z]
Out[9]=	3 5 2 2 4 a a z 2 z 3 5 7 2 z 2 z a - -- - -- + -- + --- + 4 a z + 9 a z + 5 a z - a z + 6 z - -- + ---- + z z 3 a 4 2 a a a 3 3 2 2 4 2 6 2 6 z 8 z 3 3 3 5 3 > 6 a z + 2 a z - a z - ---- - ---- - 13 a z - 21 a z - 8 a z + 3 a a 4 4 5 7 3 4 2 z 9 z 2 4 4 4 6 4 9 z > 2 a z - 24 z + ---- - ---- - 27 a z - 10 a z + 4 a z + ---- + 4 2 3 a a a 5 6 6 10 z 5 3 5 5 5 7 5 6 z 15 z > ----- + 7 a z + 16 a z + 9 a z - a z + 36 z - -- + ----- + a 4 2 a a 7 7 2 6 4 6 6 6 4 z 5 z 7 5 7 8 > 35 a z + 12 a z - 3 a z - ---- + ---- + 15 a z - 6 a z - 13 z - 3 a a 8 9 7 z 2 8 4 8 6 z 9 3 9 10 2 10 > ---- - 14 a z - 8 a z - ---- - 12 a z - 6 a z - 2 z - 2 a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	12 1 2 1 6 3 9 5 11 12 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 9 11 11 2 2 2 4 2 4 3 > ----- + ---- + ---- + 9 t + 11 q t + 6 q t + 9 q t + 3 q t + 4 2 4 2 q t q t q t 6 3 6 4 8 4 10 5 > 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a334

L11a333 L11a335