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

L11n440 L11n442

Knotscape

This page is passe. Go here instead! The 4-Component Link L11n441 Visit L11n441's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X2536 X18,11,19,12 X10,3,11,4 X4,9,1,10 X14,7,15,8 X8,13,5,14 X19,13,20,22 X15,21,16,20 X21,17,22,16 X12,17,9,18

Gauss Code:

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

Jones Polynomial:

- q-17/2 + q-15/2 - 6q-13/2 + 6q-11/2 - 12q-9/2 + 9q-7/2 - 11q-5/2 + 9q-3/2 - 6q-1/2 + 3q1/2

A2 (sl(3)) Invariant:

q-28 + 4q-26 + 7q-24 + 12q-22 + 18q-20 + 16q-18 + 18q-16 + 13q-14 + 6q-12 + 4q-10 - 4q-8 - 2q-6 - 5q-4 - 4q-2 - 3q2

HOMFLY-PT Polynomial:

az-3 + 4az-1 + 6az + 3az3 - 5a3z-3 - 17a3z-1 - 22a3z - 13a3z3 - 3a3z5 + 9a5z-3 + 23a5z-1 + 20a5z + 6a5z3 - 7a7z-3 - 11a7z-1 - 4a7z + 2a9z-3 + a9z-1

Kauffman Polynomial:

- z-2 + 4 - 6z2 + az-3 - 2az-1 + 2az - 3az3 - 3az5 - 7a2z-2 + 24a2 - 34a2z2 + 25a2z4 - 10a2z6 + 5a3z-3 - 12a3z-1 + 16a3z - 27a3z3 + 29a3z5 - 11a3z7 - 18a4z-2 + 58a4 - 75a4z2 + 44a4z4 - a4z6 - 5a4z8 + 9a5z-3 - 24a5z-1 + 37a5z - 51a5z3 + 47a5z5 - 12a5z7 - a5z9 - 19a6z-2 + 60a6 - 73a6z2 + 29a6z4 + 10a6z6 - 6a6z8 + 7a7z-3 - 23a7z-1 + 39a7z - 41a7z3 + 21a7z5 - 2a7z7 - a7z9 - 7a8z-2 + 23a8 - 26a8z2 + 10a8z4 + a8z6 - a8z8 + 2a9z-3 - 9a9z-1 + 16a9z - 14a9z3 + 6a9z5 - a9z7

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 j = 2                   3 j = 0                 3   j = -2               7 4   j = -4             4 3 1   j = -6           5 7       j = -8         7 4         j = -10       5 11           j = -12     1 1             j = -14     5               j = -16 1 1                 j = -18 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, NonAlternating, 441]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 441]]
Out[4]=	PD[X[6, 1, 7, 2], X[2, 5, 3, 6], X[18, 11, 19, 12], X[10, 3, 11, 4], > X[4, 9, 1, 10], X[14, 7, 15, 8], X[8, 13, 5, 14], X[19, 13, 20, 22], > X[15, 21, 16, 20], X[21, 17, 22, 16], X[12, 17, 9, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -11}, > {7, -6, -9, 10, 11, -3, -8, 9, -10, 8}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) -(15/2) 6 6 12 9 11 9 6 -q + q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q > 3 Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 4 7 12 18 16 18 13 6 4 4 2 5 q + --- + --- + --- + --- + --- + --- + --- + --- + --- - -- - -- - -- - 26 24 22 20 18 16 14 12 10 8 6 4 q q q q q q q q q q q q 4 2 > -- - 3 q 2 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 441]][a, z]
Out[8]=	3 5 7 9 3 5 7 9 a 5 a 9 a 7 a 2 a 4 a 17 a 23 a 11 a a -- - ---- + ---- - ---- + ---- + --- - ----- + ----- - ----- + -- + 6 a z - 3 3 3 3 3 z z z z z z z z z z 3 5 7 3 3 3 5 3 3 5 > 22 a z + 20 a z - 4 a z + 3 a z - 13 a z + 6 a z - 3 a z
In[9]:=	Kauffman[Link[11, NonAlternating, 441]][a, z]
Out[9]=	3 5 7 9 2 4 6 8 a 5 a 9 a 7 a 2 a -2 4 + 24 a + 58 a + 60 a + 23 a + -- + ---- + ---- + ---- + ---- - z - 3 3 3 3 3 z z z z z 2 4 6 8 3 5 7 9 7 a 18 a 19 a 7 a 2 a 12 a 24 a 23 a 9 a > ---- - ----- - ----- - ---- - --- - ----- - ----- - ----- - ---- + 2 a z + 2 2 2 2 z z z z z z z z z 3 5 7 9 2 2 2 4 2 > 16 a z + 37 a z + 39 a z + 16 a z - 6 z - 34 a z - 75 a z - 6 2 8 2 3 3 3 5 3 7 3 9 3 > 73 a z - 26 a z - 3 a z - 27 a z - 51 a z - 41 a z - 14 a z + 2 4 4 4 6 4 8 4 5 3 5 5 5 > 25 a z + 44 a z + 29 a z + 10 a z - 3 a z + 29 a z + 47 a z + 7 5 9 5 2 6 4 6 6 6 8 6 3 7 > 21 a z + 6 a z - 10 a z - a z + 10 a z + a z - 11 a z - 5 7 7 7 9 7 4 8 6 8 8 8 5 9 7 9 > 12 a z - 2 a z - a z - 5 a z - 6 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 4 1 1 1 5 1 1 5 3 + q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 18 8 16 8 16 7 14 6 12 6 12 5 10 5 q q t q t q t q t q t q t q t 11 7 4 5 7 4 3 7 2 > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + 3 q t 10 4 8 4 8 3 6 3 6 2 4 2 4 2 q t q t q t q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n441

L11n440 L11n442