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

L10a40 L10a42

Knotscape

This page is passe. Go here instead! The 2-Component Link L10a41 Visit L10a41's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

- q-18 - q-16 - 2q-12 + q-10 + 3q-8 + q-6 + 4q-4 + q-2 + 2 + q2 - q4 + 2q6 - q8 + q12 - q14

HOMFLY-PT Polynomial:

a-3z + a-3z3 - a-1z - 2a-1z3 - a-1z5 - 2az-1 - 3az - 2az3 - az5 + 3a3z-1 + 4a3z + 2a3z3 - a5z-1 - a5z

Kauffman Polynomial:

- 2a-4z2 + 3a-4z4 - a-4z6 + 2a-3z - 8a-3z3 + 10a-3z5 - 3a-3z7 - a-2z2 - 3a-2z4 + 8a-2z6 - 3a-2z8 + 2a-1z - 12a-1z3 + 15a-1z5 - 3a-1z7 - a-1z9 + 2z2 - 8z4 + 12z6 - 5z8 + 2az-1 - 5az + 5az5 - 2az7 - az9 - 3a2 + 4a2z2 - 2a2z4 + a2z6 - 2a2z8 + 3a3z-1 - 7a3z + 7a3z3 - 2a3z5 - 2a3z7 - 3a4 + 5a4z2 - a4z4 - 2a4z6 + a5z-1 - 2a5z + 3a5z3 - 2a5z5 - a6 + 2a6z2 - a6z4

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 j = 10                     1 j = 8                   2   j = 6                 3 1   j = 4               4 2     j = 2             4 3       j = 0           6 4         j = -2         4 6           j = -4       3 4             j = -6     1 4               j = -8   1 3                 j = -10   1                   j = -12 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[10, Alternating, 41]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 41]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[16, 10, 17, 9], X[14, 12, 15, 11], > X[10, 16, 11, 15], X[20, 17, 5, 18], X[18, 7, 19, 8], X[8, 19, 9, 20], > X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 7, -8, 3, -5, 4, -2, 10, -4, 5, -3, 6, -7, > 8, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 2 4 7 8 10 3/2 5/2 q - ---- + ---- - ---- + ---- - ------- + 8 Sqrt[q] - 7 q + 5 q - 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 7/2 9/2 > 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 2 -10 3 -6 4 -2 2 4 6 8 2 - q - q - --- + q + -- + q + -- + q + q - q + 2 q - q + 12 8 4 q q q 12 14 > q - q
In[8]:=	HOMFLYPT[Link[10, Alternating, 41]][a, z]
Out[8]=	3 5 3 3 -2 a 3 a a z z 3 5 z 2 z 3 ---- + ---- - -- + -- - - - 3 a z + 4 a z - a z + -- - ---- - 2 a z + z z z 3 a 3 a a a 5 3 3 z 5 > 2 a z - -- - a z a
In[9]:=	Kauffman[Link[10, Alternating, 41]][a, z]
Out[9]=	3 5 2 4 6 2 a 3 a a 2 z 2 z 3 5 -3 a - 3 a - a + --- + ---- + -- + --- + --- - 5 a z - 7 a z - 2 a z + z z z 3 a a 2 2 3 3 2 2 z z 2 2 4 2 6 2 8 z 12 z 3 3 > 2 z - ---- - -- + 4 a z + 5 a z + 2 a z - ---- - ----- + 7 a z + 4 2 3 a a a a 4 4 5 5 5 3 4 3 z 3 z 2 4 4 4 6 4 10 z 15 z > 3 a z - 8 z + ---- - ---- - 2 a z - a z - a z + ----- + ----- + 4 2 3 a a a a 6 6 7 5 3 5 5 5 6 z 8 z 2 6 4 6 3 z > 5 a z - 2 a z - 2 a z + 12 z - -- + ---- + a z - 2 a z - ---- - 4 2 3 a a a 7 8 9 3 z 7 3 7 8 3 z 2 8 z 9 > ---- - 2 a z - 2 a z - 5 z - ---- - 2 a z - -- - a z a 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	6 1 1 1 3 1 4 3 4 6 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 2 12 5 10 4 8 4 8 3 6 3 6 2 4 2 4 q q t q t q t q t q t q t q t q t 4 2 2 2 4 2 4 3 6 3 6 4 > ---- + 4 t + 4 q t + 3 q t + 4 q t + 2 q t + 3 q t + q t + 2 q t 8 4 10 5 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a41

L10a40 L10a42