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

L11a354 L11a356

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a355 Visit L11a355's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-23/2 - 2q-21/2 + 3q-19/2 - 6q-17/2 + 7q-15/2 - 8q-13/2 + 8q-11/2 - 7q-9/2 + 5q-7/2 - 4q-5/2 + 2q-3/2 - q-1/2

A2 (sl(3)) Invariant:

- q-34 + 2q-26 + q-24 + 3q-22 + q-16 - q-14 + 2q-12 + q-8 + q-2

HOMFLY-PT Polynomial:

- 3a3z - 4a3z3 - a3z5 - a5z-1 - 3a5z + 2a5z3 + 4a5z5 + a5z7 + a7z-1 + 6a7z + 8a7z3 + 5a7z5 + a7z7 - 3a9z - 4a9z3 - a9z5

Kauffman Polynomial:

3a3z - 7a3z3 + 5a3z5 - a3z7 + 2a4z2 - 10a4z4 + 9a4z6 - 2a4z8 + a5z-1 - 5a5z + 5a5z3 - 8a5z5 + 8a5z7 - 2a5z9 - a6 + 3a6z2 - 5a6z4 + 3a6z6 + 2a6z8 - a6z10 + a7z-1 - 9a7z + 26a7z3 - 30a7z5 + 18a7z7 - 4a7z9 + 4a8z2 - 3a8z4 + 2a8z8 - a8z10 - 2a9z + 10a9z3 - 13a9z5 + 7a9z7 - 2a9z9 + a10z2 - 5a10z4 + 4a10z6 - 2a10z8 - 3a11z + 2a11z5 - 2a11z7 + 2a12z4 - 2a12z6 - 2a13z + 4a13z3 - 2a13z5 + 2a14z2 - a14z4

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 0                       1 j = -2                     1   j = -4                   3 1   j = -6                 3 2     j = -8               4 2       j = -10             4 3         j = -12           4 4           j = -14         3 4             j = -16       3 4               j = -18     1 4                 j = -20   1 2                   j = -22   1                     j = -24 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, 355]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 355]]
Out[4]=	PD[X[12, 1, 13, 2], X[18, 7, 19, 8], X[14, 3, 15, 4], X[16, 5, 17, 6], > X[20, 10, 21, 9], X[22, 19, 11, 20], X[4, 15, 5, 16], X[6, 17, 7, 18], > X[8, 22, 9, 21], X[2, 11, 3, 12], X[10, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 3, -7, 4, -8, 2, -9, 5, -11}, > {10, -1, 11, -3, 7, -4, 8, -2, 6, -5, 9, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(23/2) 2 3 6 7 8 8 7 5 4 q - ----- + ----- - ----- + ----- - ----- + ----- - ---- + ---- - ---- + 21/2 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 q q q q q q q q q 2 1 > ---- - ------- 3/2 Sqrt[q] q
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 2 -24 3 -16 -14 2 -8 -2 -q + --- + q + --- + q - q + --- + q + q 26 22 12 q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 355]][a, z]
Out[8]=	5 7 a a 3 5 7 9 3 3 5 3 7 3 -(--) + -- - 3 a z - 3 a z + 6 a z - 3 a z - 4 a z + 2 a z + 8 a z - z z 9 3 3 5 5 5 7 5 9 5 5 7 7 7 > 4 a z - a z + 4 a z + 5 a z - a z + a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 355]][a, z]
Out[9]=	5 7 6 a a 3 5 7 9 11 13 -a + -- + -- + 3 a z - 5 a z - 9 a z - 2 a z - 3 a z - 2 a z + z z 4 2 6 2 8 2 10 2 14 2 3 3 5 3 > 2 a z + 3 a z + 4 a z + a z + 2 a z - 7 a z + 5 a z + 7 3 9 3 13 3 4 4 6 4 8 4 10 4 > 26 a z + 10 a z + 4 a z - 10 a z - 5 a z - 3 a z - 5 a z + 12 4 14 4 3 5 5 5 7 5 9 5 11 5 > 2 a z - a z + 5 a z - 8 a z - 30 a z - 13 a z + 2 a z - 13 5 4 6 6 6 10 6 12 6 3 7 5 7 > 2 a z + 9 a z + 3 a z + 4 a z - 2 a z - a z + 8 a z + 7 7 9 7 11 7 4 8 6 8 8 8 10 8 > 18 a z + 7 a z - 2 a z - 2 a z + 2 a z + 2 a z - 2 a z - 5 9 7 9 9 9 6 10 8 10 > 2 a z - 4 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 3 1 1 1 2 1 4 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 6 4 24 9 22 8 20 8 20 7 18 7 18 6 16 6 q q q t q t q t q t q t q t q t 4 3 4 4 4 4 3 4 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 16 5 14 5 14 4 12 4 12 3 10 3 10 2 8 2 q t q t q t q t q t q t q t q t 2 3 t t 2 > ---- + ---- + -- + -- + t 8 6 4 2 q t q t q q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a355

L11a354 L11a356