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

L11n415 L11n417

Knotscape

This page is passe. Go here instead! The 3-Component Link L11n416 Visit L11n416's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X8192 X14,5,15,6 X10,3,11,4 X4,13,5,14 X2738 X6,9,1,10 X18,12,19,11 X12,18,7,17 X15,20,16,21 X19,22,20,13 X21,16,22,17

Gauss Code:

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

Jones Polynomial:

- 2q-9 + 4q-8 - 6q-7 + 9q-6 - 8q-5 + 9q-4 - 6q-3 + 5q-2 - 2q-1 + 1

A2 (sl(3)) Invariant:

- q-32 - q-30 - 3q-28 - 2q-26 + 6q-20 + 5q-18 + 7q-16 + 6q-14 + 2q-12 + 4q-10 + 2q-6 + q-4 + 1

HOMFLY-PT Polynomial:

2a2 + 3a2z2 + a2z4 + 2a4z-2 + 2a4 - a4z2 - 3a4z4 - a4z6 - 5a6z-2 - 9a6 - 7a6z2 - 4a6z4 - a6z6 + 4a8z-2 + 6a8 + 4a8z2 + a8z4 - a10z-2 - a10

Kauffman Polynomial:

- 2a2 + 5a2z2 - 4a2z4 + a2z6 + a3z + 3a3z3 - 6a3z5 + 2a3z7 - 2a4z-2 + 6a4 - 4a4z2 - 4a4z6 + 2a4z8 + 5a5z-1 - 16a5z + 19a5z3 - 14a5z5 + 2a5z7 + a5z9 - 5a6z-2 + 20a6 - 36a6z2 + 36a6z4 - 22a6z6 + 6a6z8 + 9a7z-1 - 33a7z + 43a7z3 - 23a7z5 + 4a7z7 + a7z9 - 4a8z-2 + 17a8 - 32a8z2 + 34a8z4 - 16a8z6 + 4a8z8 + 5a9z-1 - 21a9z + 30a9z3 - 15a9z5 + 4a9z7 - a10z-2 + 4a10 - 5a10z2 + 2a10z4 + a10z6 + a11z-1 - 5a11z + 3a11z3

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 1                   1 j = -1                 1   j = -3               4 1   j = -5             4 3     j = -7           5 2       j = -9         4 5         j = -11       5 4           j = -13     1 4             j = -15   3 5               j = -17   2                 j = -19 2

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, NonAlternating, 416]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 416]]
Out[4]=	PD[X[8, 1, 9, 2], X[14, 5, 15, 6], X[10, 3, 11, 4], X[4, 13, 5, 14], > X[2, 7, 3, 8], X[6, 9, 1, 10], X[18, 12, 19, 11], X[12, 18, 7, 17], > X[15, 20, 16, 21], X[19, 22, 20, 13], X[21, 16, 22, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -5, 3, -4, 2, -6}, {5, -1, 6, -3, 7, -8}, > {4, -2, -9, 11, 8, -7, -10, 9, -11, 10}]
In[6]:=	Jones[L][q]
Out[6]=	2 4 6 9 8 9 6 5 2 1 - -- + -- - -- + -- - -- + -- - -- + -- - - 9 8 7 6 5 4 3 2 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 -30 3 2 6 5 7 6 2 4 2 -4 1 - q - q - --- - --- + --- + --- + --- + --- + --- + --- + -- + q 28 26 20 18 16 14 12 10 6 q q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 416]][a, z]
Out[8]=	4 6 8 10 2 4 6 8 10 2 a 5 a 4 a a 2 2 4 2 2 a + 2 a - 9 a + 6 a - a + ---- - ---- + ---- - --- + 3 a z - a z - 2 2 2 2 z z z z 6 2 8 2 2 4 4 4 6 4 8 4 4 6 6 6 > 7 a z + 4 a z + a z - 3 a z - 4 a z + a z - a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 416]][a, z]
Out[9]=	4 6 8 10 5 7 2 4 6 8 10 2 a 5 a 4 a a 5 a 9 a -2 a + 6 a + 20 a + 17 a + 4 a - ---- - ---- - ---- - --- + ---- + ---- + 2 2 2 2 z z z z z z 9 11 5 a a 3 5 7 9 11 2 2 > ---- + --- + a z - 16 a z - 33 a z - 21 a z - 5 a z + 5 a z - z z 4 2 6 2 8 2 10 2 3 3 5 3 7 3 > 4 a z - 36 a z - 32 a z - 5 a z + 3 a z + 19 a z + 43 a z + 9 3 11 3 2 4 6 4 8 4 10 4 3 5 > 30 a z + 3 a z - 4 a z + 36 a z + 34 a z + 2 a z - 6 a z - 5 5 7 5 9 5 2 6 4 6 6 6 8 6 > 14 a z - 23 a z - 15 a z + a z - 4 a z - 22 a z - 16 a z + 10 6 3 7 5 7 7 7 9 7 4 8 6 8 > a z + 2 a z + 2 a z + 4 a z + 4 a z + 2 a z + 6 a z + 8 8 5 9 7 9 > 4 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 4 2 2 3 5 1 4 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 19 7 17 6 15 6 15 5 13 5 13 4 11 4 q q q t q t q t q t q t q t q t 4 4 5 5 2 4 t t 2 > ------ + ----- + ----- + ----- + ---- + ---- + -- + - + q t 11 3 9 3 9 2 7 2 7 5 3 q q t q t q t q t q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n416

L11n415 L11n417