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

L11a175 L11a177

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a176 Visit L11a176's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X8192 X18,9,19,10 X20,13,21,14 X10,4,11,3 X14,6,15,5 X16,8,17,7 X22,16,7,15 X4,12,5,11 X12,19,13,20 X6,21,1,22 X2,18,3,17

Gauss Code:

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

Jones Polynomial:

q-9/2 - 4q-7/2 + 9q-5/2 - 15q-3/2 + 20q-1/2 - 24q1/2 + 23q3/2 - 21q5/2 + 15q7/2 - 9q9/2 + 4q11/2 - q13/2

A2 (sl(3)) Invariant:

- q-12 + 2q-10 - 3q-8 + 3q-6 - q-4 - q-2 + 5 - 4q2 + 7q4 - 2q6 + 3q8 + 2q10 - 3q12 + 3q14 - 2q16 + q18

HOMFLY-PT Polynomial:

- a-3z-1 - 2a-3z - 5a-3z3 - 4a-3z5 - a-3z7 + a-1z-1 + 5a-1z + 12a-1z3 + 13a-1z5 + 6a-1z7 + a-1z9 - 2az - 5az3 - 4az5 - az7

Kauffman Polynomial:

a-7z3 - a-7z5 - a-6z2 + 5a-6z4 - 4a-6z6 + a-5z - 6a-5z3 + 12a-5z5 - 8a-5z7 + a-4z2 - 8a-4z4 + 15a-4z6 - 10a-4z8 + a-3z-1 - 2a-3z - a-3z5 + 8a-3z7 - 8a-3z9 - a-2 + 4a-2z2 - 19a-2z4 + 27a-2z6 - 10a-2z8 - 3a-2z10 + a-1z-1 - 6a-1z + 18a-1z3 - 29a-1z5 + 32a-1z7 - 15a-1z9 + 4z2 - 16z4 + 24z6 - 7z8 - 3z10 - 3az + 6az3 - 6az5 + 12az7 - 7az9 + a2z2 - 8a2z4 + 15a2z6 - 7a2z8 - 5a3z3 + 9a3z5 - 4a3z7 - a4z2 + 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                     3   j = 10                   6 1   j = 8                 9 3     j = 6               12 6       j = 4             12 10         j = 2           12 11           j = 0         9 13             j = -2       6 11               j = -4     3 9                 j = -6   1 6                   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, 176]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 176]]
Out[4]=	PD[X[8, 1, 9, 2], X[18, 9, 19, 10], X[20, 13, 21, 14], X[10, 4, 11, 3], > X[14, 6, 15, 5], X[16, 8, 17, 7], X[22, 16, 7, 15], X[4, 12, 5, 11], > X[12, 19, 13, 20], X[6, 21, 1, 22], X[2, 18, 3, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 4, -8, 5, -10}, > {6, -1, 2, -4, 8, -9, 3, -5, 7, -6, 11, -2, 9, -3, 10, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 4 9 15 20 3/2 5/2 q - ---- + ---- - ---- + ------- - 24 Sqrt[q] + 23 q - 21 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 13/2 > 15 q - 9 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 2 3 3 -4 -2 2 4 6 8 10 5 - q + --- - -- + -- - q - q - 4 q + 7 q - 2 q + 3 q + 2 q - 10 8 6 q q q 12 14 16 18 > 3 q + 3 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 176]][a, z]
Out[8]=	3 3 5 5 1 1 2 z 5 z 5 z 12 z 3 4 z 13 z -(----) + --- - --- + --- - 2 a z - ---- + ----- - 5 a z - ---- + ----- - 3 a z 3 a 3 a 3 a a z a a a 7 7 9 5 z 6 z 7 z > 4 a z - -- + ---- - a z + -- 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 176]][a, z]
Out[9]=	2 2 2 -2 1 1 z 2 z 6 z 2 z z 4 z 2 2 -a + ---- + --- + -- - --- - --- - 3 a z + 4 z - -- + -- + ---- + a z - 3 a z 5 3 a 6 4 2 a z a a a a a 3 3 3 4 4 4 2 z 6 z 18 z 3 3 3 4 5 z 8 z > a z + -- - ---- + ----- + 6 a z - 5 a z - 16 z + ---- - ---- - 7 5 a 6 4 a a a a 4 5 5 5 5 19 z 2 4 4 4 z 12 z z 29 z 5 3 5 > ----- - 8 a z + 2 a z - -- + ----- - -- - ----- - 6 a z + 9 a z + 2 7 5 3 a a a a a 6 6 6 7 7 7 6 4 z 15 z 27 z 2 6 4 6 8 z 8 z 32 z > 24 z - ---- + ----- + ----- + 15 a z - a z - ---- + ---- + ----- + 6 4 2 5 3 a a a a a a 8 8 9 9 7 3 7 8 10 z 10 z 2 8 8 z 15 z > 12 a z - 4 a z - 7 z - ----- - ----- - 7 a z - ---- - ----- - 4 2 3 a a a a 10 9 10 3 z > 7 a z - 3 z - ----- 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 3 1 6 3 9 6 9 13 + 12 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t q t q t q t q t q t q t q t 11 2 4 4 2 6 2 6 3 8 3 > ---- + 11 q t + 12 q t + 10 q t + 12 q t + 6 q t + 9 q t + 2 q t 8 4 10 4 10 5 12 5 14 6 > 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a176

L11a175 L11a177