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

L11a501 L11a503

Knotscape

This page is passe. Go here instead! The 3-Component Link L11a502 Visit L11a502's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X8192 X10,3,11,4 X20,15,21,16 X14,5,15,6 X4,13,5,14 X12,19,7,20 X18,11,19,12 X22,17,13,18 X16,21,17,22 X2738 X6,9,1,10

Gauss Code:

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

Jones Polynomial:

- q-14 + 3q-13 - 7q-12 + 11q-11 - 14q-10 + 17q-9 - 15q-8 + 14q-7 - 9q-6 + 6q-5 - 2q-4 + q-3

A2 (sl(3)) Invariant:

- 2q-42 - q-40 - 3q-38 - 3q-36 + 3q-34 + q-32 + 7q-30 + 6q-28 + 4q-26 + 7q-24 + 4q-20 + q-18 + 3q-14 - q-12 + q-10

HOMFLY-PT Polynomial:

2a6 + 5a6z2 + 4a6z4 + a6z6 + 2a8z-2 + 4a8 + 7a8z2 + 7a8z4 + 2a8z6 - 5a10z-2 - 10a10 - 6a10z2 + a10z4 + a10z6 + 4a12z-2 + 4a12 - a12z2 - a12z4 - a14z-2

Kauffman Polynomial:

- 2a6 + 5a6z2 - 4a6z4 + a6z6 + a7z + 2a7z3 - 5a7z5 + 2a7z7 - 2a8z-2 + 6a8 - 7a8z2 + 6a8z4 - 7a8z6 + 3a8z8 + 5a9z-1 - 16a9z + 21a9z3 - 16a9z5 + 2a9z7 + 2a9z9 - 5a10z-2 + 20a10 - 33a10z2 + 27a10z4 - 17a10z6 + 5a10z8 + a10z10 + 9a11z-1 - 33a11z + 35a11z3 - 9a11z5 - 8a11z7 + 6a11z9 - 4a12z-2 + 17a12 - 33a12z2 + 44a12z4 - 30a12z6 + 9a12z8 + a12z10 + 5a13z-1 - 21a13z + 30a13z3 - 13a13z5 - 2a13z7 + 4a13z9 - a14z-2 + 4a14 - 12a14z2 + 22a14z4 - 18a14z6 + 7a14z8 + a15z-1 - 5a15z + 12a15z3 - 14a15z5 + 6a15z7 - 5a16z4 + 3a16z6 - 2a17z3 + a17z5

Khovanov Homology:

trqj r = -11 r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -5                       1 j = -7                     2 1 j = -9                   4     j = -11                 5 2     j = -13               9 4       j = -15             7 6         j = -17           10 8           j = -19         6 9             j = -21       5 8               j = -23     2 6                 j = -25   1 5                   j = -27   2                     j = -29 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, 502]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 502]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 3, 11, 4], X[20, 15, 21, 16], X[14, 5, 15, 6], > X[4, 13, 5, 14], X[12, 19, 7, 20], X[18, 11, 19, 12], X[22, 17, 13, 18], > X[16, 21, 17, 22], X[2, 7, 3, 8], X[6, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, 7, -6}, > {5, -4, 3, -9, 8, -7, 6, -3, 9, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-14 3 7 11 14 17 15 14 9 6 2 -3 -q + --- - --- + --- - --- + -- - -- + -- - -- + -- - -- + q 13 12 11 10 9 8 7 6 5 4 q q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-2 -40 3 3 3 -32 7 6 4 7 4 -18 --- - q - --- - --- + --- + q + --- + --- + --- + --- + --- + q + 42 38 36 34 30 28 26 24 20 q q q q q q q q q 3 -12 -10 > --- - q + q 14 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 502]][a, z]
Out[8]=	8 10 12 14 6 8 10 12 2 a 5 a 4 a a 6 2 8 2 2 a + 4 a - 10 a + 4 a + ---- - ----- + ----- - --- + 5 a z + 7 a z - 2 2 2 2 z z z z 10 2 12 2 6 4 8 4 10 4 12 4 6 6 8 6 > 6 a z - a z + 4 a z + 7 a z + a z - a z + a z + 2 a z + 10 6 > a z
In[9]:=	Kauffman[Link[11, Alternating, 502]][a, z]
Out[9]=	8 10 12 14 9 6 8 10 12 14 2 a 5 a 4 a a 5 a -2 a + 6 a + 20 a + 17 a + 4 a - ---- - ----- - ----- - --- + ---- + 2 2 2 2 z z z z z 11 13 15 9 a 5 a a 7 9 11 13 15 > ----- + ----- + --- + a z - 16 a z - 33 a z - 21 a z - 5 a z + z z z 6 2 8 2 10 2 12 2 14 2 7 3 > 5 a z - 7 a z - 33 a z - 33 a z - 12 a z + 2 a z + 9 3 11 3 13 3 15 3 17 3 6 4 > 21 a z + 35 a z + 30 a z + 12 a z - 2 a z - 4 a z + 8 4 10 4 12 4 14 4 16 4 7 5 > 6 a z + 27 a z + 44 a z + 22 a z - 5 a z - 5 a z - 9 5 11 5 13 5 15 5 17 5 6 6 8 6 > 16 a z - 9 a z - 13 a z - 14 a z + a z + a z - 7 a z - 10 6 12 6 14 6 16 6 7 7 9 7 > 17 a z - 30 a z - 18 a z + 3 a z + 2 a z + 2 a z - 11 7 13 7 15 7 8 8 10 8 12 8 14 8 > 8 a z - 2 a z + 6 a z + 3 a z + 5 a z + 9 a z + 7 a z + 9 9 11 9 13 9 10 10 12 10 > 2 a z + 6 a z + 4 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-7 -5 1 2 1 5 2 6 5 q + q + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 29 11 27 10 25 10 25 9 23 9 23 8 21 8 q t q t q t q t q t q t q t 8 6 9 10 8 7 6 9 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 21 7 19 7 19 6 17 6 17 5 15 5 15 4 13 4 q t q t q t q t q t q t q t q t 4 5 2 4 2 > ------ + ------ + ------ + ----- + ---- 13 3 11 3 11 2 9 2 7 q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a502

L11a501 L11a503