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

L11a49 L11a51

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a50 Visit L11a50's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X18,7,19,8 X4,19,1,20 X12,6,13,5 X10,4,11,3 X22,14,5,13 X14,22,15,21 X20,12,21,11 X16,9,17,10 X2,16,3,15 X8,17,9,18

Gauss Code:

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

Jones Polynomial:

q-9/2 - 4q-7/2 + 7q-5/2 - 12q-3/2 + 15q-1/2 - 18q1/2 + 18q3/2 - 15q5/2 + 11q7/2 - 7q9/2 + 3q11/2 - q13/2

A2 (sl(3)) Invariant:

- q-14 + 2q-12 + q-10 - q-8 + 5q-6 + q-2 + 3 - 3q2 + q4 - 4q6 + 2q10 - 2q12 + 4q14 + q16 - q18 + q20

HOMFLY-PT Polynomial:

- a-5z-1 - a-5z - a-5z3 + 2a-3z-1 + 2a-3z + a-3z3 + a-3z5 - a-1z-1 + a-1z + 3a-1z3 + 2a-1z5 - az-1 - 2az + az5 + a3z-1 - a3z3

Kauffman Polynomial:

2a-7z3 - a-7z5 + 5a-6z4 - 3a-6z6 - a-5z-1 + 5a-5z - 12a-5z3 + 14a-5z5 - 6a-5z7 - a-4 + 8a-4z2 - 19a-4z4 + 17a-4z6 - 7a-4z8 - 2a-3z-1 + 11a-3z - 15a-3z3 + a-3z5 + 7a-3z7 - 5a-3z9 - 3a-2 + 15a-2z2 - 31a-2z4 + 22a-2z6 - 5a-2z8 - 2a-2z10 - a-1z-1 + 4a-1z + 8a-1z3 - 29a-1z5 + 27a-1z7 - 10a-1z9 - 2 + 9z2 - 19z4 + 19z6 - 4z8 - 2z10 + az-1 - 3az + 3az3 - 4az5 + 10az7 - 5az9 - a2 + 2a2z2 - 10a2z4 + 16a2z6 - 6a2z8 + a3z-1 - a3z - 6a3z3 + 11a3z5 - 4a3z7 + 2a4z4 - 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                     2   j = 10                   5 1   j = 8                 6 2     j = 6               9 5       j = 4             9 6         j = 2           9 9           j = 0         8 11             j = -2       4 7               j = -4     3 8                 j = -6   1 4                   j = -8   3                     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, 50]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 50]]
Out[4]=	PD[X[6, 1, 7, 2], X[18, 7, 19, 8], X[4, 19, 1, 20], X[12, 6, 13, 5], > X[10, 4, 11, 3], X[22, 14, 5, 13], X[14, 22, 15, 21], X[20, 12, 21, 11], > X[16, 9, 17, 10], X[2, 16, 3, 15], X[8, 17, 9, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 5, -3}, {4, -1, 2, -11, 9, -5, 8, -4, 6, -7, 10, -9, 11, -2, > 3, -8, 7, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 4 7 12 15 3/2 5/2 q - ---- + ---- - ---- + ------- - 18 Sqrt[q] + 18 q - 15 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 13/2 > 11 q - 7 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 2 -10 -8 5 -2 2 4 6 10 12 3 - q + --- + q - q + -- + q - 3 q + q - 4 q + 2 q - 2 q + 12 6 q q 14 16 18 20 > 4 q + q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 50]][a, z]
Out[8]=	3 3 3 3 1 2 1 a a z 2 z z z z 3 z 3 3 -(----) + ---- - --- - - + -- - -- + --- + - - 2 a z - -- + -- + ---- - a z + 5 3 a z z z 5 3 a 5 3 a a z a z a a a a 5 5 z 2 z 5 > -- + ---- + a z 3 a a
In[9]:=	Kauffman[Link[11, Alternating, 50]][a, z]
Out[9]=	3 -4 3 2 1 2 1 a a 5 z 11 z 4 z -2 - a - -- - a - ---- - ---- - --- + - + -- + --- + ---- + --- - 3 a z - 2 5 3 a z z z 5 3 a a a z a z a a 2 2 3 3 3 3 3 2 8 z 15 z 2 2 2 z 12 z 15 z 8 z > a z + 9 z + ---- + ----- + 2 a z + ---- - ----- - ----- + ---- + 4 2 7 5 3 a a a a a a 4 4 4 5 3 3 3 4 5 z 19 z 31 z 2 4 4 4 z > 3 a z - 6 a z - 19 z + ---- - ----- - ----- - 10 a z + 2 a z - -- + 6 4 2 7 a a a a 5 5 5 6 6 6 14 z z 29 z 5 3 5 6 3 z 17 z 22 z > ----- + -- - ----- - 4 a z + 11 a z + 19 z - ---- + ----- + ----- + 5 3 a 6 4 2 a a a a a 7 7 7 8 2 6 4 6 6 z 7 z 27 z 7 3 7 8 7 z > 16 a z - a z - ---- + ---- + ----- + 10 a z - 4 a z - 4 z - ---- - 5 3 a 4 a a a 8 9 9 10 5 z 2 8 5 z 10 z 9 10 2 z > ---- - 6 a z - ---- - ----- - 5 a z - 2 z - ----- 2 3 a 2 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 3 1 4 3 8 4 8 7 11 + 9 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 > 9 q t + 9 q t + 6 q t + 9 q t + 5 q t + 6 q t + 2 q t + 10 4 10 5 12 5 14 6 > 5 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a50

L11a49 L11a51