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

L11a152 L11a154

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a153 Visit L11a153's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X8192 X10,3,11,4 X20,16,21,15 X14,5,15,6 X4,13,5,14 X22,18,7,17 X16,22,17,21 X12,20,13,19 X18,12,19,11 X2738 X6,9,1,10

Gauss Code:

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

Jones Polynomial:

- q-11/2 + 2q-9/2 - 6q-7/2 + 9q-5/2 - 13q-3/2 + 15q-1/2 - 16q1/2 + 14q3/2 - 11q5/2 + 7q7/2 - 3q9/2 + q11/2

A2 (sl(3)) Invariant:

q-16 + 3q-12 + 3q-10 + q-8 + 5q-6 - q-4 + q-2 - 3q2 + 2q4 - 3q6 + 2q8 - 2q12 + q14 - q16

HOMFLY-PT Polynomial:

3a-3z + 3a-3z3 + a-3z5 + 2a-1z-1 - 5a-1z3 - 4a-1z5 - a-1z7 - 5az-1 - 12az - 11az3 - 5az5 - az7 + 3a3z-1 + 6a3z + 4a3z3 + a3z5

Kauffman Polynomial:

a-6z2 - a-6z4 + 2a-5z3 - 3a-5z5 + a-4 - 5a-4z2 + 7a-4z4 - 6a-4z6 + 3a-3z - 10a-3z3 + 12a-3z5 - 8a-3z7 - 3a-2z2 + 8a-2z6 - 7a-2z8 + 2a-1z-1 - 2a-1z - 8a-1z3 + 14a-1z5 - a-1z7 - 4a-1z9 - 5 + 22z2 - 35z4 + 33z6 - 10z8 - z10 + 5az-1 - 21az + 29az3 - 22az5 + 19az7 - 7az9 - 5a2 + 19a2z2 - 33a2z4 + 26a2z6 - 5a2z8 - a2z10 + 3a3z-1 - 12a3z + 17a3z3 - 16a3z5 + 11a3z7 - 3a3z9 - 6a4z4 + 7a4z6 - 2a4z8 + 4a5z - 8a5z3 + 5a5z5 - a5z7

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 = 12                       1 j = 10                     2   j = 8                   5 1   j = 6                 6 2     j = 4               8 5       j = 2             8 6         j = 0           8 9           j = -2         5 7             j = -4       4 8               j = -6     2 5                 j = -8     4                   j = -10 1 2                     j = -12 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, 153]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 153]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 3, 11, 4], X[20, 16, 21, 15], X[14, 5, 15, 6], > X[4, 13, 5, 14], X[22, 18, 7, 17], X[16, 22, 17, 21], X[12, 20, 13, 19], > X[18, 12, 19, 11], X[2, 7, 3, 8], X[6, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -5, 4, -11}, > {10, -1, 11, -2, 9, -8, 5, -4, 3, -7, 6, -9, 8, -3, 7, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 2 6 9 13 15 3/2 -q + ---- - ---- + ---- - ---- + ------- - 16 Sqrt[q] + 14 q - 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 11/2 > 11 q + 7 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 3 3 -8 5 -4 -2 2 4 6 8 12 q + --- + --- + q + -- - q + q - 3 q + 2 q - 3 q + 2 q - 2 q + 12 10 6 q q q 14 16 > q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 153]][a, z]
Out[8]=	3 3 3 2 5 a 3 a 3 z 3 3 z 5 z 3 3 3 --- - --- + ---- + --- - 12 a z + 6 a z + ---- - ---- - 11 a z + 4 a z + a z z z 3 3 a a a 5 5 7 z 4 z 5 3 5 z 7 > -- - ---- - 5 a z + a z - -- - a z 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 153]][a, z]
Out[9]=	3 -4 2 2 5 a 3 a 3 z 2 z 3 5 -5 + a - 5 a + --- + --- + ---- + --- - --- - 21 a z - 12 a z + 4 a z + a z z z 3 a a 2 2 2 3 3 3 2 z 5 z 3 z 2 2 2 z 10 z 8 z 3 > 22 z + -- - ---- - ---- + 19 a z + ---- - ----- - ---- + 29 a z + 6 4 2 5 3 a a a a a a 4 4 5 3 3 5 3 4 z 7 z 2 4 4 4 3 z > 17 a z - 8 a z - 35 z - -- + ---- - 33 a z - 6 a z - ---- + 6 4 5 a a a 5 5 6 6 12 z 14 z 5 3 5 5 5 6 6 z 8 z > ----- + ----- - 22 a z - 16 a z + 5 a z + 33 z - ---- + ---- + 3 a 4 2 a a a 7 7 2 6 4 6 8 z z 7 3 7 5 7 8 > 26 a z + 7 a z - ---- - -- + 19 a z + 11 a z - a z - 10 z - 3 a a 8 9 7 z 2 8 4 8 4 z 9 3 9 10 2 10 > ---- - 5 a z - 2 a z - ---- - 7 a z - 3 a z - z - a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 2 4 2 5 4 8 9 + 8 q + ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 12 6 10 6 10 5 8 4 6 4 6 3 4 3 4 2 q t q t q t q t q t q t q t q t 5 8 7 2 4 4 2 6 2 6 3 > ----- + - + ---- + 6 q t + 8 q t + 5 q t + 6 q t + 2 q t + 2 2 t 2 q t q t 8 3 8 4 10 4 12 5 > 5 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a153

L11a152 L11a154