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

L11a408 L11a410

Knotscape

This page is passe. Go here instead! The 3-Component Link L11a409 Visit L11a409's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-4 + 3q-3 - 7q-2 + 11q-1 - 14 + 18q - 16q2 + 16q3 - 11q4 + 7q5 - 3q6 + q7

A2 (sl(3)) Invariant:

- q-12 - q-8 - 3q-6 + q-4 - 3q-2 + 2 + 4q2 + 3q4 + 10q6 + 3q8 + 7q10 + 2q12 + 3q16 - q18 + q20

HOMFLY-PT Polynomial:

2a-4z-2 + 4a-4 + 6a-4z2 + 4a-4z4 + a-4z6 - 5a-2z-2 - 13a-2 - 19a-2z2 - 15a-2z4 - 6a-2z6 - a-2z8 + 4z-2 + 12 + 15z2 + 9z4 + 2z6 - a2z-2 - 3a2 - 3a2z2 - a2z4

Kauffman Polynomial:

- a-8z2 + a-8z4 - 2a-7z3 + 3a-7z5 - a-6 + 5a-6z2 - 7a-6z4 + 6a-6z6 + 5a-5z3 - 10a-5z5 + 8a-5z7 - 2a-4z-2 + 8a-4 - 17a-4z2 + 21a-4z4 - 18a-4z6 + 9a-4z8 + 5a-3z-1 - 18a-3z + 33a-3z3 - 26a-3z5 + a-3z7 + 5a-3z9 - 5a-2z-2 + 20a-2 - 50a-2z2 + 72a-2z4 - 58a-2z6 + 16a-2z8 + a-2z10 + 9a-1z-1 - 33a-1z + 50a-1z3 - 24a-1z5 - 12a-1z7 + 8a-1z9 - 4z-2 + 15 - 34z2 + 56z4 - 45z6 + 10z8 + z10 + 5az-1 - 19az + 30az3 - 15az5 - 4az7 + 3az9 - a2z-2 + 3a2 - 7a2z2 + 13a2z4 - 11a2z6 + 3a2z8 + a3z-1 - 4a3z + 6a3z3 - 4a3z5 + a3z7

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 = 15                       1 j = 13                     3 1 j = 11                   4     j = 9                 7 3     j = 7               9 4       j = 5             8 8         j = 3           10 8           j = 1         6 10             j = -1       5 8               j = -3     2 6                 j = -5   1 5                   j = -7   2                     j = -9 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 409]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 409]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[8, 18, 9, 17], X[14, 8, 15, 7], > X[18, 10, 19, 9], X[10, 12, 5, 11], X[22, 19, 11, 20], X[20, 15, 21, 16], > X[16, 21, 17, 22], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, -3, 5, -6}, > {6, -2, 11, -4, 8, -9, 3, -5, 7, -8, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-4 3 7 11 2 3 4 5 6 7 -14 - q + -- - -- + -- + 18 q - 16 q + 16 q - 11 q + 7 q - 3 q + q 3 2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -8 3 -4 3 2 4 6 8 10 12 2 - q - q - -- + q - -- + 4 q + 3 q + 10 q + 3 q + 7 q + 2 q + 6 2 q q 16 18 20 > 3 q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 409]][a, z]
Out[8]=	2 2 2 4 13 2 4 2 5 a 2 6 z 19 z 12 + -- - -- - 3 a + -- + ----- - ----- - -- + 15 z + ---- - ----- - 4 2 2 4 2 2 2 2 4 2 a a z a z a z z a a 4 4 6 6 8 2 2 4 4 z 15 z 2 4 6 z 6 z z > 3 a z + 9 z + ---- - ----- - a z + 2 z + -- - ---- - -- 4 2 4 2 2 a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 409]][a, z]
Out[9]=	2 3 -6 8 20 2 4 2 5 a 5 9 5 a a 15 - a + -- + -- + 3 a - -- - ----- - ----- - -- + ---- + --- + --- + -- - 4 2 2 4 2 2 2 2 3 a z z z a a z a z a z z a z 2 2 2 2 18 z 33 z 3 2 z 5 z 17 z 50 z > ---- - ---- - 19 a z - 4 a z - 34 z - -- + ---- - ----- - ----- - 3 a 8 6 4 2 a a a a a 3 3 3 3 4 2 2 2 z 5 z 33 z 50 z 3 3 3 4 z > 7 a z - ---- + ---- + ----- + ----- + 30 a z + 6 a z + 56 z + -- - 7 5 3 a 8 a a a a 4 4 4 5 5 5 5 7 z 21 z 72 z 2 4 3 z 10 z 26 z 24 z 5 > ---- + ----- + ----- + 13 a z + ---- - ----- - ----- - ----- - 15 a z - 6 4 2 7 5 3 a a a a a a a 6 6 6 7 7 7 3 5 6 6 z 18 z 58 z 2 6 8 z z 12 z > 4 a z - 45 z + ---- - ----- - ----- - 11 a z + ---- + -- - ----- - 6 4 2 5 3 a a a a a a 8 8 9 9 7 3 7 8 9 z 16 z 2 8 5 z 8 z 9 > 4 a z + a z + 10 z + ---- + ----- + 3 a z + ---- + ---- + 3 a z + 4 2 3 a a a a 10 10 z > z + --- 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 2 1 5 2 6 5 8 10 q + 10 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t q t q t q t q t q t q t q t 6 q 3 5 5 2 7 2 7 3 9 3 9 4 > --- + 8 q t + 8 q t + 8 q t + 9 q t + 4 q t + 7 q t + 3 q t + t 11 4 13 5 13 6 15 6 > 4 q t + 3 q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a409

L11a408 L11a410