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

L11a484 L11a486

Knotscape

This page is passe. Go here instead! The 3-Component Link L11a485 Visit L11a485's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X4,15,1,16 X12,6,13,5 X8493 X22,11,19,12 X18,22,5,21 X20,10,21,9 X10,17,11,18 X16,19,17,20 X2,14,3,13

Gauss Code:

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

Jones Polynomial:

q-6 - 6q-5 + 13q-4 - 20q-3 + 28q-2 - 30q-1 + 32 - 26q + 19q2 - 11q3 + 5q4 - q5

A2 (sl(3)) Invariant:

q-18 - 3q-16 - q-14 + 2q-12 - 4q-10 + 8q-8 + 2q-6 + 4q-4 + 8q-2 - 2 + 9q2 - 3q4 + 3q6 + 4q8 - 3q10 + 3q12 - q14

HOMFLY-PT Polynomial:

a-2z-2 + 2a-2 - 2a-2z4 - a-2z6 - 2z-2 - 6 - 2z2 + 5z4 + 4z6 + z8 + a2z-2 + 6a2 + 2a2z2 - 4a2z4 - 2a2z6 - 2a4 + a4z4

Kauffman Polynomial:

a-5z5 - 4a-4z4 + 5a-4z6 + 4a-3z3 - 13a-3z5 + 11a-3z7 + a-2z-2 - 4a-2 + 2a-2z2 + 8a-2z4 - 20a-2z6 + 15a-2z8 - 2a-1z-1 + 2a-1z + 12a-1z3 - 17a-1z5 - 5a-1z7 + 12a-1z9 + 2z-2 - 11 + 10z2 + 38z4 - 67z6 + 26z8 + 4z10 - 2az-1 + 6az + 10az3 - 3az5 - 36az7 + 24az9 + a2z-2 - 12a2 + 10a2z2 + 38a2z4 - 69a2z6 + 24a2z8 + 4a2z10 + 6a3z + 2a3z3 - 8a3z5 - 14a3z7 + 12a3z9 - 4a4 + 2a4z2 + 12a4z4 - 26a4z6 + 13a4z8 + 2a5z - 8a5z5 + 6a5z7 + a6z6

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 11                       1 j = 9                     4   j = 7                   7 1   j = 5                 12 4     j = 3               14 7       j = 1             18 12         j = -1           16 18           j = -3         12 14             j = -5       8 16               j = -7     5 12                 j = -9   1 8                   j = -11   5                     j = -13 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, 485]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 485]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[12, 6, 13, 5], > X[8, 4, 9, 3], X[22, 11, 19, 12], X[18, 22, 5, 21], X[20, 10, 21, 9], > X[10, 17, 11, 18], X[16, 19, 17, 20], X[2, 14, 3, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {10, -8, 7, -6}, > {4, -1, 2, -5, 8, -9, 6, -4, 11, -2, 3, -10, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-6 6 13 20 28 30 2 3 4 5 32 + q - -- + -- - -- + -- - -- - 26 q + 19 q - 11 q + 5 q - q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 3 -14 2 4 8 2 4 8 2 4 6 -2 + q - --- - q + --- - --- + -- + -- + -- + -- + 9 q - 3 q + 3 q + 16 12 10 8 6 4 2 q q q q q q q 8 10 12 14 > 4 q - 3 q + 3 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 485]][a, z]
Out[8]=	2 4 2 2 4 2 1 a 2 2 2 4 2 z -6 + -- + 6 a - 2 a - -- + ----- + -- - 2 z + 2 a z + 5 z - ---- - 2 2 2 2 2 2 a z a z z a 6 2 4 4 4 6 z 2 6 8 > 4 a z + a z + 4 z - -- - 2 a z + z 2 a
In[9]:=	Kauffman[Link[11, Alternating, 485]][a, z]
Out[9]=	2 4 2 4 2 1 a 2 2 a 2 z 3 -11 - -- - 12 a - 4 a + -- + ----- + -- - --- - --- + --- + 6 a z + 6 a z + 2 2 2 2 2 a z z a a z a z z 2 3 3 5 2 2 z 2 2 4 2 4 z 12 z 3 > 2 a z + 10 z + ---- + 10 a z + 2 a z + ---- + ----- + 10 a z + 2 3 a a a 4 4 5 5 5 3 3 4 4 z 8 z 2 4 4 4 z 13 z 17 z > 2 a z + 38 z - ---- + ---- + 38 a z + 12 a z + -- - ----- - ----- - 4 2 5 3 a a a a a 6 6 5 3 5 5 5 6 5 z 20 z 2 6 4 6 > 3 a z - 8 a z - 8 a z - 67 z + ---- - ----- - 69 a z - 26 a z + 4 2 a a 7 7 8 6 6 11 z 5 z 7 3 7 5 7 8 15 z > a z + ----- - ---- - 36 a z - 14 a z + 6 a z + 26 z + ----- + 3 a 2 a a 9 2 8 4 8 12 z 9 3 9 10 2 10 > 24 a z + 13 a z + ----- + 24 a z + 12 a z + 4 z + 4 a z a
In[10]:=	Kh[L][q, t]
Out[10]=	18 1 5 1 8 5 12 8 16 -- + 18 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 12 14 16 3 3 2 5 2 5 3 > ----- + ---- + --- + 12 q t + 14 q t + 7 q t + 12 q t + 4 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 7 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a485

L11a484 L11a486