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

L11a355 L11a357

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a356 Visit L11a356's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-15/2 - 2q-13/2 + 4q-11/2 - 8q-9/2 + 10q-7/2 - 13q-5/2 + 13q-3/2 - 12q-1/2 + 9q1/2 - 6q3/2 + 3q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-22 - q-18 + 3q-14 + 4q-10 + q-8 + 2q-4 - 2q-2 + 3 - q2 + q6 - q8 + q10

HOMFLY-PT Polynomial:

- 2a-1z - 3a-1z3 - a-1z5 - az-1 - az + 4az3 + 4az5 + az7 + a3z-1 + 6a3z + 9a3z3 + 5a3z5 + a3z7 - 4a5z - 4a5z3 - a5z5

Kauffman Polynomial:

2a-3z3 - a-3z5 - a-2z2 + 6a-2z4 - 3a-2z6 + 3a-1z - 9a-1z3 + 12a-1z5 - 5a-1z7 + 4z2 - 12z4 + 12z6 - 5z8 + az-1 - 2az - 3az3 - az5 + 4az7 - 3az9 - a2 + 6a2z2 - 16a2z4 + 11a2z6 - 3a2z8 - a2z10 + a3z-1 - 10a3z + 27a3z3 - 30a3z5 + 16a3z7 - 5a3z9 + 4a4z2 - 2a4z4 + a4z6 - a4z10 - 3a5z + 12a5z3 - 9a5z5 + 5a5z7 - 2a5z9 - a6z2 + 4a6z6 - 2a6z8 + 2a7z - 7a7z3 + 7a7z5 - 2a7z7 - 4a8z2 + 4a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                       1 j = 6                     2   j = 4                   4 1   j = 2                 5 2     j = 0               7 4       j = -2             7 6         j = -4           6 6           j = -6         5 8             j = -8       3 5               j = -10     1 5                 j = -12   1 3                   j = -14   1                     j = -16 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, 356]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 356]]
Out[4]=	PD[X[12, 1, 13, 2], X[16, 5, 17, 6], X[14, 3, 15, 4], X[20, 10, 21, 9], > X[18, 8, 19, 7], X[22, 17, 11, 18], X[8, 20, 9, 19], X[4, 15, 5, 16], > X[6, 22, 7, 21], X[2, 11, 3, 12], X[10, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 3, -8, 2, -9, 5, -7, 4, -11}, > {10, -1, 11, -3, 8, -2, 6, -5, 7, -4, 9, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 2 4 8 10 13 13 12 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 9 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 7/2 > 6 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -18 3 4 -8 2 2 2 6 8 10 3 - q - q + --- + --- + q + -- - -- - q + q - q + q 14 10 4 2 q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 356]][a, z]
Out[8]=	3 3 a a 2 z 3 5 3 z 3 3 3 5 3 -(-) + -- - --- - a z + 6 a z - 4 a z - ---- + 4 a z + 9 a z - 4 a z - z z a a 5 z 5 3 5 5 5 7 3 7 > -- + 4 a z + 5 a z - a z + a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 356]][a, z]
Out[9]=	3 2 2 a a 3 z 3 5 7 2 z 2 2 -a + - + -- + --- - 2 a z - 10 a z - 3 a z + 2 a z + 4 z - -- + 6 a z + z z a 2 a 3 3 4 2 6 2 8 2 2 z 9 z 3 3 3 5 3 > 4 a z - a z - 4 a z + ---- - ---- - 3 a z + 27 a z + 12 a z - 3 a a 4 5 5 7 3 4 6 z 2 4 4 4 8 4 z 12 z 5 > 7 a z - 12 z + ---- - 16 a z - 2 a z + 4 a z - -- + ----- - a z - 2 3 a a a 6 3 5 5 5 7 5 6 3 z 2 6 4 6 6 6 > 30 a z - 9 a z + 7 a z + 12 z - ---- + 11 a z + a z + 4 a z - 2 a 7 8 6 5 z 7 3 7 5 7 7 7 8 2 8 > a z - ---- + 4 a z + 16 a z + 5 a z - 2 a z - 5 z - 3 a z - a 6 8 9 3 9 5 9 2 10 4 10 > 2 a z - 3 a z - 5 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	6 1 1 1 3 1 5 3 5 7 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 5 8 6 6 7 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 4 t + 5 q t + 2 q t + 4 q t + 6 3 6 2 4 2 4 2 q t q t q t q t q t 4 3 6 3 8 4 > q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a356

L11a355 L11a357