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

L11a301 L11a303

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a302 Visit L11a302's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 2q-5/2 - 4q-3/2 + 6q-1/2 - 9q1/2 + 10q3/2 - 10q5/2 + 8q7/2 - 7q9/2 + 4q11/2 - 2q13/2 + q15/2

A2 (sl(3)) Invariant:

q-10 + q-6 + q-4 + 3 - q2 + q4 + 3q10 + 2q14 - q18 - q22

HOMFLY-PT Polynomial:

4a-5z + 4a-5z3 + a-5z5 - 4a-3z - 8a-3z3 - 5a-3z5 - a-3z7 - a-1z-1 - 4a-1z - 8a-1z3 - 5a-1z5 - a-1z7 + az-1 + 4az + 4az3 + az5

Kauffman Polynomial:

- 4a-8z2 + 4a-8z4 - a-8z6 + a-7z - 6a-7z3 + 7a-7z5 - 2a-7z7 + a-6z2 - 2a-6z4 + 5a-6z6 - 2a-6z8 - 5a-5z + 18a-5z3 - 14a-5z5 + 7a-5z7 - 2a-5z9 + 2a-4z2 + 2a-4z4 - 3a-4z6 + 2a-4z8 - a-4z10 - 6a-3z + 23a-3z3 - 30a-3z5 + 16a-3z7 - 4a-3z9 - 7a-2z4 + 2a-2z6 + a-2z8 - a-2z10 - a-1z-1 + 6a-1z - 13a-1z3 + a-1z5 + 4a-1z7 - 2a-1z9 + 1 + 2z2 - 10z4 + 9z6 - 3z8 - az-1 + 5az - 9az3 + 9az5 - 3az7 - a2z2 + 5a2z4 - 2a2z6 - a3z + 3a3z3 - a3z5

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 16                       1 j = 14                     1   j = 12                   3 1   j = 10                 4 1     j = 8               4 3       j = 6             6 4         j = 4           4 4           j = 2         5 6             j = 0       3 6               j = -2     1 3                 j = -4   1 3                   j = -6   1                     j = -8 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, 302]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 302]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[4, 9, 5, 10], > X[16, 6, 17, 5], X[22, 14, 9, 13], X[20, 16, 21, 15], X[14, 22, 15, 21], > X[18, 8, 19, 7], X[6, 18, 7, 17], X[8, 20, 1, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -4, 5, -10, 9, -11}, > {4, -1, 2, -3, 6, -8, 7, -5, 10, -9, 11, -7, 8, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 2 4 6 3/2 5/2 7/2 -q + ---- - ---- + ------- - 9 Sqrt[q] + 10 q - 10 q + 8 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 13/2 15/2 > 7 q + 4 q - 2 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 -6 -4 2 4 10 14 18 22 3 + q + q + q - q + q + 3 q + 2 q - q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 302]][a, z]
Out[8]=	3 3 3 5 1 a 4 z 4 z 4 z 4 z 8 z 8 z 3 z -(---) + - + --- - --- - --- + 4 a z + ---- - ---- - ---- + 4 a z + -- - a z z 5 3 a 5 3 a 5 a a a a a 5 5 7 7 5 z 5 z 5 z z > ---- - ---- + a z - -- - -- 3 a 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 302]][a, z]
Out[9]=	2 2 2 1 a z 5 z 6 z 6 z 3 2 4 z z 2 z 1 - --- - - + -- - --- - --- + --- + 5 a z - a z + 2 z - ---- + -- + ---- - a z z 7 5 3 a 8 6 4 a a a a a a 3 3 3 3 4 2 2 6 z 18 z 23 z 13 z 3 3 3 4 4 z > a z - ---- + ----- + ----- - ----- - 9 a z + 3 a z - 10 z + ---- - 7 5 3 a 8 a a a a 4 4 4 5 5 5 5 2 z 2 z 7 z 2 4 7 z 14 z 30 z z 5 3 5 > ---- + ---- - ---- + 5 a z + ---- - ----- - ----- + -- + 9 a z - a z + 6 4 2 7 5 3 a a a a a a a 6 6 6 6 7 7 7 7 6 z 5 z 3 z 2 z 2 6 2 z 7 z 16 z 4 z > 9 z - -- + ---- - ---- + ---- - 2 a z - ---- + ---- + ----- + ---- - 8 6 4 2 7 5 3 a a a a a a a a 8 8 8 9 9 9 10 10 7 8 2 z 2 z z 2 z 4 z 2 z z z > 3 a z - 3 z - ---- + ---- + -- - ---- - ---- - ---- - --- - --- 6 4 2 5 3 a 4 2 a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 3 1 3 3 2 4 6 + 5 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 6 q t + 4 q t + 8 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t 4 2 6 2 6 3 8 3 8 4 10 4 10 5 > 4 q t + 6 q t + 4 q t + 4 q t + 3 q t + 4 q t + q t + 12 5 12 6 14 6 16 7 > 3 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a302

L11a301 L11a303