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

L11a523 L11a525

Knotscape

This page is passe. Go here instead! The 3-Component Link L11a524 Visit L11a524's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-5 + 5q-4 - 11q-3 + 18q-2 - 24q-1 + 28 - 26q + 24q2 - 16q3 + 10q4 - 4q5 + q6

A2 (sl(3)) Invariant:

- q-14 + 3q-12 - 3q-10 + 3q-8 + q-6 - 4q-4 + 6q-2 - 4 + 7q2 + 3q4 + 4q6 + 9q8 - q10 + 4q12 - q16 + q18

HOMFLY-PT Polynomial:

a-4z-2 + a-4 + 2a-4z2 + a-4z4 - 2a-2z-2 - 2a-2 - 5a-2z2 - 6a-2z4 - 2a-2z6 + z-2 + z2 + 5z4 + 4z6 + z8 + a2 - 2a2z4 - a2z6

Kauffman Polynomial:

a-6z2 - 2a-6z4 + a-6z6 + 4a-5z3 - 8a-5z5 + 4a-5z7 - a-4z-2 + 4a-4 - 10a-4z2 + 16a-4z4 - 18a-4z6 + 8a-4z8 + 2a-3z-1 - 5a-3z + 6a-3z3 - a-3z5 - 11a-3z7 + 8a-3z9 - 2a-2z-2 + 6a-2 - 22a-2z2 + 50a-2z4 - 52a-2z6 + 16a-2z8 + 3a-2z10 + 2a-1z-1 - 5a-1z + 11a-1z3 + 2a-1z5 - 28a-1z7 + 18a-1z9 - z-2 + 2 - 11z2 + 42z4 - 57z6 + 22z8 + 3z10 + 13az3 - 20az5 - 2az7 + 10az9 - a2 + 6a2z4 - 19a2z6 + 14a2z8 + 4a3z3 - 14a3z5 + 11a3z7 - 4a4z4 + 5a4z6 + a5z5

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 = 13                       1 j = 11                     3   j = 9                   7 1   j = 7                 10 4     j = 5               14 6       j = 3             13 11         j = 1           15 13           j = -1         10 14             j = -3       8 14               j = -5     4 11                 j = -7   1 7                   j = -9   4                     j = -11 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, 524]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 524]]
Out[4]=	PD[X[8, 1, 9, 2], X[22, 10, 13, 9], X[18, 8, 19, 7], X[20, 17, 21, 18], > X[6, 22, 1, 21], X[4, 11, 5, 12], X[16, 6, 17, 5], X[10, 16, 11, 15], > X[12, 20, 7, 19], X[2, 13, 3, 14], X[14, 3, 15, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 11, -6, 7, -5}, {3, -1, 2, -8, 6, -9}, > {10, -11, 8, -7, 4, -3, 9, -4, 5, -2}]
In[6]:=	Jones[L][q]
Out[6]=	-5 5 11 18 24 2 3 4 5 6 28 - q + -- - -- + -- - -- - 26 q + 24 q - 16 q + 10 q - 4 q + q 4 3 2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 3 3 3 -6 4 6 2 4 6 8 10 -4 - q + --- - --- + -- + q - -- + -- + 7 q + 3 q + 4 q + 9 q - q + 12 10 8 4 2 q q q q q 12 16 18 > 4 q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 524]][a, z]
Out[8]=	2 2 4 4 -4 2 2 -2 1 2 2 2 z 5 z 4 z 6 z a - -- + a + z + ----- - ----- + z + ---- - ---- + 5 z + -- - ---- - 2 4 2 2 2 4 2 4 2 a a z a z a a a a 6 2 4 6 2 z 2 6 8 > 2 a z + 4 z - ---- - a z + z 2 a
In[9]:=	Kauffman[Link[11, Alternating, 524]][a, z]
Out[9]=	2 4 6 2 -2 1 2 2 2 5 z 5 z 2 z 2 + -- + -- - a - z - ----- - ----- + ---- + --- - --- - --- - 11 z + -- - 4 2 4 2 2 2 3 a z 3 a 6 a a a z a z a z a a 2 2 3 3 3 4 10 z 22 z 4 z 6 z 11 z 3 3 3 4 2 z > ----- - ----- + ---- + ---- + ----- + 13 a z + 4 a z + 42 z - ---- + 4 2 5 3 a 6 a a a a a 4 4 5 5 5 16 z 50 z 2 4 4 4 8 z z 2 z 5 3 5 > ----- + ----- + 6 a z - 4 a z - ---- - -- + ---- - 20 a z - 14 a z + 4 2 5 3 a a a a a 6 6 6 7 7 5 5 6 z 18 z 52 z 2 6 4 6 4 z 11 z > a z - 57 z + -- - ----- - ----- - 19 a z + 5 a z + ---- - ----- - 6 4 2 5 3 a a a a a 7 8 8 9 28 z 7 3 7 8 8 z 16 z 2 8 8 z > ----- - 2 a z + 11 a z + 22 z + ---- + ----- + 14 a z + ---- + a 4 2 3 a a a 9 10 18 z 9 10 3 z > ----- + 10 a z + 3 z + ----- a 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	14 1 4 1 7 4 11 8 14 -- + 15 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t 10 3 3 2 5 2 5 3 7 3 > --- + 13 q t + 13 q t + 11 q t + 14 q t + 6 q t + 10 q t + q t 7 4 9 4 9 5 11 5 13 6 > 4 q t + 7 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a524

L11a523 L11a525