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

L11a26 L11a28

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a27 Visit L11a27's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X16,13,17,14 X14,7,15,8 X8,15,9,16 X20,11,21,12 X22,18,5,17 X18,22,19,21 X12,19,13,20 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-21/2 - 3q-19/2 + 7q-17/2 - 12q-15/2 + 16q-13/2 - 19q-11/2 + 18q-9/2 - 17q-7/2 + 12q-5/2 - 7q-3/2 + 3q-1/2 - q1/2

A2 (sl(3)) Invariant:

- q-34 - 2q-32 + q-30 - 2q-26 + 5q-24 + 3q-18 - 2q-16 + 4q-14 - q-12 + 2q-10 + 3q-8 - 4q-6 + 3q-4 - 1 + q2

HOMFLY-PT Polynomial:

- az - az3 - a3z-1 - a3z + a3z5 + 2a5z-1 + 3a5z + 3a5z3 + 2a5z5 - 3a7z-1 - 7a7z - 5a7z3 + 3a9z-1 + 4a9z - a11z-1

Kauffman Polynomial:

- az + 2az3 - az5 - 2a2z2 + 5a2z4 - 3a2z6 - a3z-1 + 3a3z - 4a3z3 + 7a3z5 - 5a3z7 + a4z2 - 5a4z4 + 8a4z6 - 6a4z8 - 2a5z-1 + 11a5z - 20a5z3 + 15a5z5 - 2a5z7 - 4a5z9 - 2a6 + 11a6z2 - 27a6z4 + 29a6z6 - 12a6z8 - a6z10 - 3a7z-1 + 17a7z - 31a7z3 + 21a7z5 + 3a7z7 - 7a7z9 + 5a8z2 - 17a8z4 + 25a8z6 - 10a8z8 - a8z10 - 3a9z-1 + 13a9z - 24a9z3 + 22a9z5 - 3a9z7 - 3a9z9 + 2a10 - 6a10z2 + 3a10z4 + 6a10z6 - 4a10z8 - a11z-1 + 3a11z - 7a11z3 + 8a11z5 - 3a11z7 + a12 - 3a12z2 + 3a12z4 - a12z6

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 = 2                       1 j = 0                     2   j = -2                   5 1   j = -4                 8 3     j = -6               9 4       j = -8             9 8         j = -10           10 9           j = -12         7 10             j = -14       5 9               j = -16     2 7                 j = -18   1 5                   j = -20   2                     j = -22 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, 27]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 27]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[16, 13, 17, 14], X[14, 7, 15, 8], > X[8, 15, 9, 16], X[20, 11, 21, 12], X[22, 18, 5, 17], X[18, 22, 19, 21], > X[12, 19, 13, 20], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, -5, 11, -2, 6, -9, 3, -4, 5, -3, 7, -8, > 9, -6, 8, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 3 7 12 16 19 18 17 12 7 q - ----- + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q q 3 > ------- - Sqrt[q] Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 2 -30 2 5 3 2 4 -12 2 3 4 -1 - q - --- + q - --- + --- + --- - --- + --- - q + --- + -- - -- + 32 26 24 18 16 14 10 8 6 q q q q q q q q q 3 2 > -- + q 4 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 27]][a, z]
Out[8]=	3 5 7 9 11 a 2 a 3 a 3 a a 3 5 7 9 -(--) + ---- - ---- + ---- - --- - a z - a z + 3 a z - 7 a z + 4 a z - z z z z z 3 5 3 7 3 3 5 5 5 > a z + 3 a z - 5 a z + a z + 2 a z
In[9]:=	Kauffman[Link[11, Alternating, 27]][a, z]
Out[9]=	3 5 7 9 11 6 10 12 a 2 a 3 a 3 a a 3 5 -2 a + 2 a + a - -- - ---- - ---- - ---- - --- - a z + 3 a z + 11 a z + z z z z z 7 9 11 2 2 4 2 6 2 8 2 > 17 a z + 13 a z + 3 a z - 2 a z + a z + 11 a z + 5 a z - 10 2 12 2 3 3 3 5 3 7 3 9 3 > 6 a z - 3 a z + 2 a z - 4 a z - 20 a z - 31 a z - 24 a z - 11 3 2 4 4 4 6 4 8 4 10 4 12 4 > 7 a z + 5 a z - 5 a z - 27 a z - 17 a z + 3 a z + 3 a z - 5 3 5 5 5 7 5 9 5 11 5 2 6 > a z + 7 a z + 15 a z + 21 a z + 22 a z + 8 a z - 3 a z + 4 6 6 6 8 6 10 6 12 6 3 7 5 7 > 8 a z + 29 a z + 25 a z + 6 a z - a z - 5 a z - 2 a z + 7 7 9 7 11 7 4 8 6 8 8 8 10 8 > 3 a z - 3 a z - 3 a z - 6 a z - 12 a z - 10 a z - 4 a z - 5 9 7 9 9 9 6 10 8 10 > 4 a z - 7 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 5 1 2 1 5 2 7 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 22 9 20 8 18 8 18 7 16 7 16 6 14 6 q q q t q t q t q t q t q t q t 9 7 10 10 9 9 8 9 4 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 14 5 12 5 12 4 10 4 10 3 8 3 8 2 6 2 6 q t q t q t q t q t q t q t q t q t 8 t 2 2 > ---- + 2 t + -- + q t 4 2 q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a27

L11a26 L11a28