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

L11a382 L11a384

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a383 Visit L11a383's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-9/2 - 2q-7/2 + 4q-5/2 - 7q-3/2 + 8q-1/2 - 11q1/2 + 10q3/2 - 9q5/2 + 7q7/2 - 4q9/2 + 2q11/2 - q13/2

A2 (sl(3)) Invariant:

- q-12 - 2q-8 + q-6 + q-4 + 2q-2 + 5 + 3q4 - 2q6 - q12 + 2q14 + q18

HOMFLY-PT Polynomial:

- a-3z-1 - 8a-3z - 12a-3z3 - 6a-3z5 - a-3z7 + a-1z-1 + 17a-1z + 32a-1z3 + 24a-1z5 + 8a-1z7 + a-1z9 - 8az - 12az3 - 6az5 - az7

Kauffman Polynomial:

- a-7z + 3a-7z3 - a-7z5 - a-6z2 + 5a-6z4 - 2a-6z6 + a-5z - 5a-5z3 + 8a-5z5 - 3a-5z7 + 10a-4z2 - 22a-4z4 + 15a-4z6 - 4a-4z8 + a-3z-1 - 7a-3z + 11a-3z3 - 18a-3z5 + 11a-3z7 - 3a-3z9 - a-2 + 16a-2z2 - 31a-2z4 + 14a-2z6 - a-2z8 - a-2z10 + a-1z-1 - 19a-1z + 47a-1z3 - 49a-1z5 + 23a-1z7 - 5a-1z9 + 10z2 - 10z4 + 3z6 + z8 - z10 - 9az + 22az3 - 15az5 + 7az7 - 2az9 + a2z2 - 2a2z4 + 5a2z6 - 2a2z8 + a3z - 6a3z3 + 7a3z5 - 2a3z7 - 4a4z2 + 4a4z4 - a4z6

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 = 14                       1 j = 12                     1   j = 10                   3 1   j = 8                 4 1     j = 6               5 3       j = 4             6 5         j = 2           5 4           j = 0         4 7             j = -2       3 4               j = -4     1 4                 j = -6   1 3                   j = -8   1                     j = -10 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, 383]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 383]]
Out[4]=	PD[X[12, 1, 13, 2], X[2, 13, 3, 14], X[14, 3, 15, 4], X[10, 11, 1, 12], > X[22, 15, 11, 16], X[16, 8, 17, 7], X[6, 22, 7, 21], X[20, 6, 21, 5], > X[4, 20, 5, 19], X[18, 10, 19, 9], X[8, 18, 9, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -9, 8, -7, 6, -11, 10, -4}, > {4, -1, 2, -3, 5, -6, 11, -10, 9, -8, 7, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 2 4 7 8 3/2 5/2 q - ---- + ---- - ---- + ------- - 11 Sqrt[q] + 10 q - 9 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 13/2 > 7 q - 4 q + 2 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 2 -6 -4 2 4 6 12 14 18 5 - q - -- + q + q + -- + 3 q - 2 q - q + 2 q + q 8 2 q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 383]][a, z]
Out[8]=	3 3 5 5 1 1 8 z 17 z 12 z 32 z 3 6 z 24 z -(----) + --- - --- + ---- - 8 a z - ----- + ----- - 12 a z - ---- + ----- - 3 a z 3 a 3 a 3 a a z a a a 7 7 9 5 z 8 z 7 z > 6 a z - -- + ---- - a z + -- 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 383]][a, z]
Out[9]=	2 2 -2 1 1 z z 7 z 19 z 3 2 z 10 z -a + ---- + --- - -- + -- - --- - ---- - 9 a z + a z + 10 z - -- + ----- + 3 a z 7 5 3 a 6 4 a z a a a a a 2 3 3 3 3 16 z 2 2 4 2 3 z 5 z 11 z 47 z 3 3 3 > ----- + a z - 4 a z + ---- - ---- + ----- + ----- + 22 a z - 6 a z - 2 7 5 3 a a a a a 4 4 4 5 5 5 4 5 z 22 z 31 z 2 4 4 4 z 8 z 18 z > 10 z + ---- - ----- - ----- - 2 a z + 4 a z - -- + ---- - ----- - 6 4 2 7 5 3 a a a a a a 5 6 6 6 49 z 5 3 5 6 2 z 15 z 14 z 2 6 4 6 > ----- - 15 a z + 7 a z + 3 z - ---- + ----- + ----- + 5 a z - a z - a 6 4 2 a a a 7 7 7 8 8 9 3 z 11 z 23 z 7 3 7 8 4 z z 2 8 3 z > ---- + ----- + ----- + 7 a z - 2 a z + z - ---- - -- - 2 a z - ---- - 5 3 a 4 2 3 a a a a a 9 10 5 z 9 10 z > ---- - 2 a z - z - --- a 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 3 1 4 3 4 4 7 + 5 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + ---- + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t q t q t 2 4 4 2 6 2 6 3 8 3 8 4 > 4 q t + 6 q t + 5 q t + 5 q t + 3 q t + 4 q t + q t + 10 4 10 5 12 5 14 6 > 3 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a383

L11a382 L11a384