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

L11a535 L11a537

Knotscape

This page is passe. Go here instead! The 4-Component Link L11a536 Visit L11a536's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X2536 X18,12,19,11 X10,3,11,4 X4,9,1,10 X16,7,17,8 X8,15,5,16 X22,18,15,17 X12,22,13,21 X20,14,21,13 X14,20,9,19

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 2q-11/2 - 6q-9/2 + 9q-7/2 - 15q-5/2 + 15q-3/2 - 18q-1/2 + 14q1/2 - 12q3/2 + 7q5/2 - 4q7/2 + q9/2

A2 (sl(3)) Invariant:

q-22 + 2q-20 + 2q-16 + 5q-14 + q-12 + 8q-10 + 11q-8 + 9q-6 + 14q-4 + 8q-2 + 10 + 5q2 + q4 + 5q6 - 2q8 + 2q12 - q14

HOMFLY-PT Polynomial:

a-3z3 - a-1z-3 - 3a-1z-1 - 3a-1z - a-1z3 - a-1z5 + 3az-3 + 7az-1 + 6az + 2az3 - az5 - 3a3z-3 - 4a3z-1 + 3a3z3 + a5z-3 - a5z-1 - 3a5z + a7z-1

Kauffman Polynomial:

2a-4z4 - a-4z6 - 7a-3z3 + 11a-3z5 - 4a-3z7 - 10a-2z4 + 16a-2z6 - 6a-2z8 + a-1z-3 - 6a-1z-1 + 13a-1z - 14a-1z3 + 9a-1z5 + 4a-1z7 - 4a-1z9 - 3z-2 + 13 - 17z2 - 2z4 + 19z6 - 8z8 - z10 + 3az-3 - 14az-1 + 28az - 29az3 + 13az5 + 5az7 - 6az9 - 6a2z-2 + 24a2 - 33a2z2 + 17a2z4 + 3a2z6 - 5a2z8 - a2z10 + 3a3z-3 - 12a3z-1 + 21a3z - 26a3z3 + 19a3z5 - 6a3z7 - 2a3z9 - 3a4z-2 + 11a4 - 16a4z2 + 10a4z4 - a4z6 - 3a4z8 + a5z-3 - 3a5z-1 + 3a5z - a5z3 + 3a5z5 - 3a5z7 - a6 + 3a6z4 - 2a6z6 + a7z-1 - 3a7z + 3a7z3 - a7z5

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 = 10                       1 j = 8                     3   j = 6                   4 1   j = 4                 8 3     j = 2               9 7       j = 0             9 5         j = -2           7 10           j = -4         8 8             j = -6       2 8               j = -8     4 7                 j = -10   1 5                   j = -12   1                     j = -14 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]=	4
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 536]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 536]]
Out[4]=	PD[X[6, 1, 7, 2], X[2, 5, 3, 6], X[18, 12, 19, 11], X[10, 3, 11, 4], > X[4, 9, 1, 10], X[16, 7, 17, 8], X[8, 15, 5, 16], X[22, 18, 15, 17], > X[12, 22, 13, 21], X[20, 14, 21, 13], X[14, 20, 9, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -9, 10, -11}, > {7, -6, 8, -3, 11, -10, 9, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 2 6 9 15 15 18 -q + ----- - ---- + ---- - ---- + ---- - ------- + 14 Sqrt[q] - 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 3/2 5/2 7/2 9/2 > 12 q + 7 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 2 2 5 -12 8 11 9 14 8 2 4 10 + q + --- + --- + --- + q + --- + -- + -- + -- + -- + 5 q + q + 20 16 14 10 8 6 4 2 q q q q q q q q 6 8 12 14 > 5 q - 2 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 536]][a, z]
Out[8]=	3 5 3 5 7 1 3 a 3 a a 3 7 a 4 a a a 3 z 5 -(----) + --- - ---- + -- - --- + --- - ---- - -- + -- - --- + 6 a z - 3 a z + 3 3 3 3 a z z z z z a a z z z z 3 3 5 z z 3 3 3 z 5 > -- - -- + 2 a z + 3 a z - -- - a z 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 536]][a, z]
Out[9]=	3 5 2 4 2 4 6 1 3 a 3 a a 3 6 a 3 a 6 13 + 24 a + 11 a - a + ---- + --- + ---- + -- - -- - ---- - ---- - --- - 3 3 3 3 2 2 2 a z a z z z z z z z 3 5 7 14 a 12 a 3 a a 13 z 3 5 7 > ---- - ----- - ---- + -- + ---- + 28 a z + 21 a z + 3 a z - 3 a z - z z z z a 3 3 2 2 2 4 2 7 z 14 z 3 3 3 5 3 > 17 z - 33 a z - 16 a z - ---- - ----- - 29 a z - 26 a z - a z + 3 a a 4 4 5 7 3 4 2 z 10 z 2 4 4 4 6 4 11 z > 3 a z - 2 z + ---- - ----- + 17 a z + 10 a z + 3 a z + ----- + 4 2 3 a a a 5 6 6 9 z 5 3 5 5 5 7 5 6 z 16 z > ---- + 13 a z + 19 a z + 3 a z - a z + 19 z - -- + ----- + a 4 2 a a 7 7 2 6 4 6 6 6 4 z 4 z 7 3 7 5 7 > 3 a z - a z - 2 a z - ---- + ---- + 5 a z - 6 a z - 3 a z - 3 a a 8 9 8 6 z 2 8 4 8 4 z 9 3 9 10 2 10 > 8 z - ---- - 5 a z - 3 a z - ---- - 6 a z - 2 a z - z - a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	10 1 1 1 5 4 7 2 8 9 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 8 8 7 2 2 2 4 2 4 3 > ----- + ---- + ---- + 5 t + 9 q t + 7 q t + 8 q t + 3 q t + 4 2 4 2 q t q t q t 6 3 6 4 8 4 10 5 > 4 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a536

L11a535 L11a537