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

L11a280 L11a282

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a281 Visit L11a281's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-15/2 - 5q-13/2 + 11q-11/2 - 18q-9/2 + 23q-7/2 - 27q-5/2 + 26q-3/2 - 23q-1/2 + 16q1/2 - 9q3/2 + 4q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-22 + 3q-20 - 2q-18 + q-16 + 4q-14 - 4q-12 + 6q-10 - q-8 + 2q-6 + 3q-4 - 4q-2 + 5 - 4q2 + 2q6 - 2q8 + q10

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a-3z3 - a-3z5 - a-2z2 + 5a-2z4 - 4a-2z6 + a-1z - 5a-1z3 + 11a-1z5 - 8a-1z7 + 2z2 - 11z4 + 17z6 - 11z8 + az - 3az3 - 5az5 + 13az7 - 10az9 + 7a2z2 - 33a2z4 + 39a2z6 - 12a2z8 - 4a2z10 - a3z-1 + 2a3z + 7a3z3 - 30a3z5 + 43a3z7 - 20a3z9 + a4 + 5a4z2 - 28a4z4 + 40a4z6 - 11a4z8 - 4a4z10 - a5z-1 + a5z + a5z3 - 4a5z5 + 17a5z7 - 10a5z9 + a6z2 - 10a6z4 + 21a6z6 - 10a6z8 - a7z - 3a7z3 + 9a7z5 - 5a7z7 + a8z4 - 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                     3   j = 4                   6 1   j = 2                 10 3     j = 0               13 6       j = -2             14 11         j = -4           13 12           j = -6         10 14             j = -8       8 13               j = -10     4 11                 j = -12   1 7                   j = -14   4                     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, 281]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 281]]
Out[4]=	PD[X[10, 1, 11, 2], X[8, 9, 1, 10], X[12, 4, 13, 3], X[22, 16, 9, 15], > X[2, 17, 3, 18], X[4, 22, 5, 21], X[14, 5, 15, 6], X[20, 13, 21, 14], > X[16, 12, 17, 11], X[6, 19, 7, 20], X[18, 7, 19, 8]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -5, 3, -6, 7, -10, 11, -2}, > {2, -1, 9, -3, 8, -7, 4, -9, 5, -11, 10, -8, 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 5 11 18 23 27 26 23 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 16 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 > 9 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 3 2 -16 4 4 6 -8 2 3 4 2 5 - q + --- - --- + q + --- - --- + --- - q + -- + -- - -- - 4 q + 20 18 14 12 10 6 4 2 q q q q q q q q 6 8 10 > 2 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 281]][a, z]
Out[8]=	3 5 3 5 a a z 3 5 2 z 3 3 3 5 3 z -(--) + -- - - + 2 a z - 4 a z + a z - ---- + 4 a z - a z - a z - -- + z z a a a 5 3 5 5 5 7 3 7 > 3 a z + 2 a z - a z + a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 281]][a, z]
Out[9]=	3 5 2 4 a a z 3 5 7 2 z 2 2 4 2 a - -- - -- + - + a z + 2 a z + a z - a z + 2 z - -- + 7 a z + 5 a z + z z a 2 a 3 3 4 6 2 z 5 z 3 3 3 5 3 7 3 4 5 z > a z + -- - ---- - 3 a z + 7 a z + a z - 3 a z - 11 z + ---- - 3 a 2 a a 5 5 2 4 4 4 6 4 8 4 z 11 z 5 3 5 > 33 a z - 28 a z - 10 a z + a z - -- + ----- - 5 a z - 30 a z - 3 a a 6 5 5 7 5 6 4 z 2 6 4 6 6 6 8 6 > 4 a z + 9 a z + 17 z - ---- + 39 a z + 40 a z + 21 a z - a z - 2 a 7 8 z 7 3 7 5 7 7 7 8 2 8 > ---- + 13 a z + 43 a z + 17 a z - 5 a z - 11 z - 12 a z - a 4 8 6 8 9 3 9 5 9 2 10 4 10 > 11 a z - 10 a z - 10 a z - 20 a z - 10 a z - 4 a z - 4 a z
In[10]:=	Kh[L][q, t]
Out[10]=	11 1 4 1 7 4 11 8 13 13 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 10 14 13 12 14 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 6 t + 10 q t + 3 q t + 6 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 + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a281

L11a280 L11a282